math 1010-90 online 2 3 2 5 6 · 2002. 12. 31. · x 2 x 3 x 2 x 4 expand x2 5x 6 x2 2x 8 x2 5x 6...

Math 1010-90 online

Fall 2002

Answers for Exam 1

Tom RobbinsThis set contains no questions, just the answers to

Exam 1. You can look at the individual problems, ordownload a hard copy with all the answers as usual.Let me know if you have any questions!

1.(0 pts) Express the following as a fraction. Re-duce any common factors:

43� 5

6 � 86� 5

6 � 136

43� 5

6 � 86 � 5

6 � 36 � 1

2

43� 5

6 � 4 � 53 � 6 � 2 � 5

3 � 3 � 109

43� 5

6 � 43� 6

5 � 85

2.(0 pts)The Number System

Give an example of� a natural number: 1, 2, 3, 4, ....� an integer that’s not a natural number: 0, -1,-2,� a rational number that’s not an integer: 1/2,3/17, -17/19� a real number that’s not rational: π,

�2

3.(0 pts) LanguageConvert the following phrases to algebraic expres-sions:� The quotient of a and b is a

b .� The product of a and the sum of c and d isa � c � d .� The difference of the product of a and b andthe ratio of a and b is ab � a

b .

4.(0 pts) Simplification of Algebraic ExpressionsWrite the expression below in the form Ax2 � Bx � Cwhere A, B and C are real numbers.

� x � 2 � x � 3 � x2 � 5x � 6 �(Hence A � 1, B � 5, and C � 6. Writing down

those values is a handy way to design WeBWorKproblems, but it was not necessary on the exam.)

5.(0 pts)Precedence. Let x � 8, y � 4 and z � 2. Compute

the following (and express the answer as an integeror fraction):

x � y � y � z � 8 � 4 � 4 � 2 � 8 � 1 � 2 � 5� x � y � y � z � � 8 � 4 � 4 � 2 � 4 � 4 � 2 � 1 � 2 � 1x � y �� y � z � 8 � 4 �� 4 � 2 � 8 � 4 � 2 � 8 � 2 � 6

x � y � z � 8 � 4 � 2 � 2 � 2 � 1� x � y � z � � 8 � 4 �� 2 � 2 � 2 � 1x �� y � z � 8 � 4 � 2 � 8 � 2 � 4

6.(0 pts) A Linear EquationSolve the equation

3x � 6 � 3

We proceed as usual:

3x � 6 � 3 � � 63x � � 3 � � 3

x � � 1 � the answer

7.(0 pts) Another Linear EquationSolve the equation

x3 � 2x

5 �� 1

This is actually a straightforward linear equation,but the fractions may be confusing. To get rid of themwe multiply with the common denominator (15) onboth sides.

x3 � 2x

5 � � 1 � � 155x � 6x � � 15 � simplify� x � � 15 � � � � 1

x � 15 � the answer

8.(0 pts) Yet another linear equation:Solve the equation� x � 2 � x � 3 � � x � 2 � x � 4 ��The trick here is to expand on both sides and then

cancel the nonlinear term x2 on both sides.1

� x � 2 �� x � 3 � � x � 2 �� x � 4 � expandx2 � 5x � 6 � x2 � 2x � 8 � � x2

5x � 6 � 2x � 8 � � 2x � 63x � � 14 � � 3

x � � 143 � the answer

9.(0 pts) One More Linear Equation.Solve the Equation

12x � 2 � 1

4 � x�

We obtain1

2x � 2 � 14 � x � take reciprocals

2x � 2 � 4 � x � � 2 � x3x � 6 � � 2

x � 2 � the answer

10.(0 pts) A word ProblemThe “invoice price” of a car you consider buying is

$20,000. The dealer offers you the car for “5% over

invoice”. When you balk he offers to split a “$1,000factory incentive” evenly with you, on top of the pricehe just offered. What’s the price you would pay if youaccept the deal?

Some students protested that they did not knowwhat an invoice price is, and they never bought a car.You can answer this question without that knowledgeand experience. However, for your info, the ”invoiceprice” is supposed to be the amount of money thedealer pays the car maker. It’s essentially public in-formation that you can look up in various places onthe web or in the appropriate literature. The situationis murky, however, since dealers and car makers haveways to hide additional profits from the consumer.

Whatever the invoice price means, it is $20,000.The dealer offers you the car for 5% more than that.5% of $20,000 is $1,000. So if you stopped here theprice of the car would be $21,000. Splitting $1,000evenly (whatever ”factory incentive” means) reducesthat price by $500. So your total price would be$20,500.

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c�

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Math 1010-90 online

Answers for Exam 2

Tom RobbinsThis set contains no questions, just the answers to Exam 2. You can look at the individual problems, or

download a hard copy with all the answers as usual. Let me know if you have any questions!1.(0 pts) Solving Linear Equations. Solve the equation

x � 1x � 2 � x � 3

x � 4

To get rid of the denominators we multiply with both of them on both sides. Then we eliminate the paren-theses using the Distributive Law, cancel the term x2, and solve the resulting linear equation as usual:

x � 1x � 2 � x � 3

x � 4 � � � x � 1 �� x � 4 � x � 1 � x � 4 � � x � 3 � x � 2 � Distributex2 � 3x � 4 � x2 � x � 6 � � x2

3x � 4 � � x � 6 � � x � 44x � � 2 � � 4x � � 1

2 � the solutionOf course we check our answer:

� 12 � 1� 12� 2 � � 3

232�� 1 � � 7

272� � 1

2 � 3� 12� 4�

It works!

2.(0 pts) Solving Inequalities. Solve the inequal-ity

2x � 1 � 3x � 2 � ��We process inequalities like equalities except

that we reverse the inequality when multiplying witha negative factor on both sides of the inequality. Sowe obtain:

2x � 1 � 3x � 2 � � 3x � 1� x � � 3 � � � � 1 x � 3 � the answer

To check the answer we note that we get equality in�� when x � 3 and that the right side of �� increasesfaster than the left side as x increases.

3.(0 pts) Solving Absolute Values Equations.Solve the equation� x � 1 � � � x � 2 ��

The key to solving absolute value equations isthe fact that the argument of an absolute value may bepositive or negative. For the given problem there aretwo possibilities (after eliminating equivalent equa-tions obtained by multiplying with � 1 on both sides):

x � 1 � x � 2and

x � 1 �� x � 2 �� x � 2In the first case there is no solution. In the second

case the solution of the linear equation is

x � 12�

That’s the only solution of �� .1

To check the answer we substitute that value of xin the original equation �� and see that the equationis indeed satisfied.

4.(0 pts) Graphs of linear Equations. Draw thegraph of the equation

x � 2y � 3 � 0

This graph is easier to draw if we solve the givenequation for y, obtaining

y �� x2 � 3

2�

The graph of this linear equation is a straight line ofslope � 1

2 and y-intercept � 32 , as shown here:

5.(0 pts) Graphs of Linear Equations. Draw thegraph of the line with slope 1

2 that passes through thepoint � 1 � 1 and give the slope intercept form of itsequation.

Since the slope is 12 the slope intercept form of its

equation is

y � 12

x � b

where b needs to be determined. We know that thepoint 1 � 1 lies on the graph, and so we have the equa-tion

1 � 12� 1 � b

which tells us that

b � 12�

The required equation is therefore given by

y � 12

x � 12�

Its graph is shown in this Figure:

6.(0 pts) More Graphs.Draw the graphs of the two equations

y � x and y � � x2� 3

2and indicate graphically and algebraically where thetwo lines intersect.

The graph of the second equation is a straight linewith slope � 1

2 and y-intercept 32 and the graph of the

first equation is a straight line with slope 1 throughthe origin. They are shown (in red and green, respec-tively) in this Figure:

They appear to intersect in the point � 1 � 1 and wecan check immediately that x � y � 1 does indeedsatisfy both equations.

7.(0 pts) Functions. Let

f � x � 2x � 1

Compute� f � 2 � 2 � 2 � 1 � 5 �� f � x � 1 � 2 � � x � 1 � 1 � 2x � 3� f � f � x � 2 � � 2x � 1 � 1 � 4x � 32

8.(0 pts)Powers and Radicals.Write 27 � 2

3 as a fraction.We obtain

27 � 23 � ! 27

13 " � 2 � 3 � 2 � 1

32 � 19�

9.(0 pts) Complex Numbers.Let u � 1 � i and v � 1 � i. Express as complex

numbers in standard form:� u � v � 1 � i � 1 � i � 2,� u � v � 1 � i � � 1 � i � 2i� u � v � � 1 � i � 1 � i � 12 � i2 � 1 � 1 � 2 �� uv � 1 � i

1 � i � � 1 � i � 1 � i � 1 � i � 1 � i � 12 � 2i � i2

12 � i2 �2i2 � i

10.(0 pts) Why basket ball players weigh somuch. Two siblings look exactly alike, except thatone is five feet tall and the other seven feet tall. Thusevery linear dimension (lengths of arms, diameter ofhead, torso, etc.) of the taller person equals 7

5 timesthe corresponding dimension of the shorter person.The shorter person weighs 125 pounds. How muchdoes the taller person weigh?

Multiplying linear dimensions with a certain factormultiplies the volume and the weight by the cube ofthat factor. The weight of the larger person is there-fore

#75 $ 3 � 125 � 343

125� 125 � 343 pounds �


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U of U Math 1010 Online

Tom Robbins.

WeBWorK assignment number 1.

due 9/2/02 at 11:59 PM.The main purpose of this first WeBWorK set is to

review some prerequisites for this class and to helpyou familiarize yourself with WeBWorK.

Here are some hints on how to use WeBWorK ef-fectively:� After first logging into WeBWorK change

your password.� Find out how to print a hard copy on the com-puter system that you are going to use. Con-tact me if you have any problems. Print a hardcopy of this assignment. Note, however, thatthe online versions of these problems mayhave links to the web pages of this course thatare absent on the hard copy.� Get to work on this set right away and answerthese questions well before the deadline. Notonly will this give you the chance to figureout what’s wrong if an answer is not accepted,you also will avoid the likely rush and con-gestion prior to the deadline.� The primary purpose of the WeBWorK as-signments in this class is to give you the op-portunity to learn by having instant feedbackon your active solution of relevant problems.Make the best of it!

Procrastination is hazardous!Peter Alfeld, JWB 127 or 236, 581-6842.1.(10 pts) This first question is just an exercise in

entering answers into WeBWorK. It also gives youan opportunity to experiment with entering differentarithmetic and algebraic expressions into WeB-WorK and seeing what WeBWorK really thinks youare doing (as opposed to what you believe it shouldthink).

Notice the buttons on this page and try them out be-fore moving to the next problem. Use the ”Back”Button on your browser to get back here whenneeded.

� ”Prob. List” gets you back to the list of allproblems in this set.� ”Next” gets you to the next question in thisset.� ”Submit Answer” submits your answer asyou might expect, but there may be otherways to do so. Specifically, in this problem,there is only one question. In that case youcan submit your answer by typing it into theanswer window and then pressing ”Return”(or ”Enter”) on your keyboard. But even inthis case, you can also type the answer andclick on the ”submit” button. There is noharm in submitting an answer even if you arenot quite sure that it’s correct, since if it is notyou have an unlimited number of additionaltries. On the other hand, it is usually moreefficient to print your own private problemsset, work out the answers in a quiet environ-ment like your home, and then sit down infront of a computer and enter your answers. Ifsome are wrong you can try to fix them rightat the computer, or you may want to go backand work on them quietly elsewhere beforereturning to the computer.� Pressing on the ”Preview Answer” Buttonmakes WeBWorK display what it thinks youentered in the answer window. After using”Preview” you can modify your answer anduse a ”Preview Again” button.� ”typeset” denotes the ordinary display modeon your workstation, but ”formatted text” is alittle faster. The option of plain ”text” is notuseful.� ”Logout” terminates this WeBWorK sessionfor you. You can of course log back in andcontinue.� ”Feedback” enables you to send a message toyour instructor (Peter Alfeld), and the WeB-WorK assistants. If you use this way of send-ing e-mail the recipients receive information

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about your WeBWorK state, in addition toyour actual message.� The ”Help” Button transports you to an of-ficial WeBWorK help page that has a moreinformation than this first problem.� ”Problem Sets” transports you back to thepage where you can select a certain problemset. When you do this particular problem inthis first set, there is only one set, but eventu-ally there will be 13 of them.� For all problems in this course you will beable to see the Answers to the problems af-ter the due date. Go to a problem, clickon ”show correct answers”, and then click on”submit answer”. You can also download andprint a hard copy with the answers showing.These answers are the precise strings againstwhich WeBWorK compares your answer. Ifthe answer is an algebraic expression your an-swer needs to be equivalent to the WeBWorKanswer, but it may be in a different form. Forexample if WeBWorK thinks the answer is2 � a, it is OK for you to type a � a instead. IfWeBWorK expects a numerical answer thenyou can usually enter it as an arithmetic ex-pression (like 1 � 7 instead of � 142857), andusually WeBWorK will expect your answerto be within one tenth of one percent of whatit thinks the answer is.� Most of the problems (including this one) inthis course will also have solutions attachedthat you can see after the due date by click-ing on ”show solutions” followed by ”sub-mit answers”. The solutions are text typedby your instructor that gives more informa-tion than the ”answers”, and in particular of-ten explains how the answers can be obtained.

Now for the meat of this problem. Notice that the an-swer window is extra large so you can try the thingssuggested above.Type the number 3 here:

.Try entering other expressions and use the pre-view button to see what WeBWorK thinks you en-tered. Return to this problem to try out thingswhen you get stuck somewhere else.

Here are some good examples to try. Check themall out using the Preview button. (In later questionson this set you will get to use what you learn here.)Never mind that you may have already answered thecorrect answer 3. Once you get credit for an answerit won’t be taken away by trying other answers.a/2b versus a/2/b versus a/(2b)a/b+c versus a/(b+c)a+b**2 versus (a+b)**2sqrt a+b versus sqrt(a+b)4/3 pi r**2 versus (4/3) pi r**2 (In other words, ifyou are not sure use parentheses freely.)

Note: WeBWorK will not usually let you enter al-gebraic expressions when the answer is a number, andit will only let you use certain variables when the an-swer is in fact an algebraic expression. So the abovewindow, and the opportunity for experimentationthat it offers is unique. Make good use of it!Presumably this has been your first encounter withWeBWorK. Come back here to try things out andto refresh your memory if you get stuck somewheredown the line.

2.(10 pts) The purpose of this exercise is to illus-trate further the use of the buttons on this page andto show you the most common way in which WeB-WorK processes partially correct problems. Try en-tering incorrect answers in the answer fields below,to see what happens. (This time WeBWorK will re-ject algebraic expressions since I told it to expect anumerical answer.)Type the number 4 here: .Type the number 5 here: .

3.(10 pts)In the first few problems, now that you are famil-

iar with the basic mechanics of WeBWorK, you willbe asked to evaluate some arithmetic expressions andenter the answer as a number into WeBWorK. Youmay of course use a calculator. In later problems youwill be able to enter the answer as an arithmetic ex-pression, but at present your answer must be a num-ber such as 4, -4, or 17.5.Evaluate the expression5 � 9 � 9 = .(Remember that by convention a missing arithmetic operator means multiplication.)

4.(10 pts)2

Evaluate the expression4 � 1 � 1 = .

5.(10 pts)Evaluate the expression2 �� 7 � 3 = .Enter you answer as a decimal number listing at least4 decimal digits. (WeBWorK will reject your answerif it differs by more than one tenth of 1 percent fromwhat it thinks the answer is.)

6.(10 pts)Evaluate the expression15 � � 6 � 5 = .

7.(10 pts)Evaluate the expression5 � � 9 � 10 = .

8.(10 pts) This problem illustrates the standardrules of arithmetic precedence:� Multiplication and Division precede Subtrac-

tion and Addition.� Among operations with the same level ofprecedence, evaluation proceeds from left toright.� However, expressions in parentheses are eval-uated first.

Evaluate the expression6 � 5 � 3 � 8 =Evaluate the expression6 � � 5 � 3 � 8 =Evaluate the expression6 � � 5 � 3 � 8 =

9.(10 pts) This problem provides more illustrationsof the use of parentheses.Evaluate the expression2 � 4 � 3 � 2 =Evaluate the expression2 � � 4 � 3 � 2 =Evaluate the expression2 � � 4 � 3 � 2 =Evaluate the expression2 � � 4 � � 3 � 2 =

10.(10 pts) The key idea in Algebra is to use vari-ables in addition to numbers. Sometimes we needto replace variables with specific numbers. That’s

called evaluating an algebraic expression . For ex-ample, if a � 2 then 3a � 6, and we say that we eval-uated the expression 3a at a � 2. We’ll do this sortof thing all semester long, and in this problem youget your first experience with evaluating algebraic ex-pressions. Again, the emphasis in these exercises inon understanding the rules of arithmetic precedence.

Let a � 13 � b � 5 � c � 13 �Then a � b � c �and � a � b � c �As usual, enter your answers as decimal numbers

with at least 4 digits.11.(10 pts) Let r � 3 �Then 4 � π � r �and 4 �� π � r �12.(10 pts) Let a � 3 � b � 5 � c � 7 � d � 8 �Then a � b � c � d � ,� a � b �� c � d � ,

a � � b � c � d � , anda � b �� c � d � .

13.(10 pts) The next three problems are like thepreceding three, except that you need to get all an-swers correct before WeBWorK will give you credit.This will be true for many problems in this class. Thepurpose of insisting on all answers being correct is toencourage you to think about the whole context of theproblem rather than the individual pieces.Let a � 2 � 5 � b � 4 � 9 � c � 6 � 1 �

Then a � b � c �and � a � b � c �14.(10 pts) Let r � 4 � 7 �Then 4 � π � r �and 4 �� π � r �15.(10 pts) Let a � 2 � 5 � b � 4 � 3 � c �

5 � 9 � d � 6 � 9 �Then

a � b � c � d � ,� a � b �� c � d � ,a � � b � c � d � , anda � b �� c � d � .

16.(10 pts) In this and the following prob-lems you will practice entering algebraicexpressions into WeBWorK. Remember theRules of Arithmetic Precedence and use paren-theses freely to make your meaning clear. Most of

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the difficulties students have with WeBWorK are dueto not appreciating the precise rules that govern theinterpretation of what you enter. This is not just amatter of WeBWorK understanding what you are try-ing to say, The rules are used universally all overthe world. Appreciating and applying them properlyis also crucial, for example, in computer program-ming. Make sure you understand what’s going inthese problems. If you enter a wrong expression usethe Preview Button to see what WeBWorK thinks youhave entered.

We start simply. Enter here the expressiona � b.

17.(10 pts) Enter here the expressiona � 12 � b

Enter here the expression

a � bc � d

If WeBWorK rejects your answer use the previewbutton to see what it thinks you are trying to tell it.

18.(10 pts) Enter here the expression1

1a� 1

b


a � b � 11 � 1

a � b

19.(10 pts)Enter here the expression

ab� c

def� g

h�

20.(10 pts)The square x2 of a number x simply means the prod-uct of x with itself. For example, 32 � 3 � 3 � 9. Youcan enter a number such as 32 as 3**2. (An expres-sion such as 32 or x2 is called a power. We will learna great deal more about powers during this semester.)Enter here the expression x2

The square root�

x of a number x is a numberwhose square equals x. For example

�25 � 5 since

52 � 5 � 5 � 25.

To enter square roots you can use the function sqrt.For example, to enter the square root of 2 you cantype sqrt(2).

Enter here the expression�

a21.(10 pts) Enter here the expression�

a � b

Enter here the expressiona�

a � bEnter here the expression

a � b�a � b

22.(10 pts)Enter here the expression%

x2 � y2


x%

x2 � y2

Enter here the expressionx � y%x2 � y2

23.(10 pts)Enter here the expression� b � � b2 � 4ac

2aNote: this is an expression that gives the solution ofa quadratic equation by the quadratic formula. Wewill learn much more about it later in the semester.

24.(10 pts) Consider the following expressions:

A � a � bc

and

B � a � bc

For each of the WeBWorK phrases below write Aif they define A and B if they define B.You need to get all answers correct before obtainingcredit.

a � b � c (This is the standard way to enter A, soenter A).

4

� a � b � c (This is the standard way to enter B,so enter B). � � a � b � c

a � � b � c � a � � b � c 25.(10 pts) Consider again the formula for the so-

lution of a quadratic equation:

x � � b � � b2 � 4ac2a

For each of the WeBWorK phrases below enter aT (true) if the phrase describes x, correctly, and a F(false) otherwise.You need to get all answers correct before obtainingcredit.

-b+sqrt(b**2-4*a*c)/2a(-b+sqrt(b**2-4*a*c))/2a

(-b+sqrt(b**2-4*a*c))/(2a)26.(10 pts) More of the same.

(-b+sqrt(b**2-(4*a*c)))/(2a)(-b+(sqrt(b**2-4*a*c)))/(2a)((-b+sqrt(b**2-4*a*c))/(2a))(-b+(sqrt(b**2-4*a*c))/(2a))

(-b+sqrt(b*b-(4*a*c)))/(2a)27.(10 pts)For each of the WeBWorK phrases below enter a

T (true) if the two given phrases describe the samealgebraic expression and an F (false) otherwise. Oneway you can decide whether the phrases are equiva-lent is to substitute specific values for a, b, etc. If youget two different results the two phrases are certainlynot equivalent. If you get the same values there issmall chance this happened accidentally for just thatchoice of particular values. In any case, pay closeattention to when these phrases are equivalent andwhen they are not, it will help you tremendously withfuture WeBWorK assignments.a � b b � aa � b � c a � � b � c a � b � c a � � b � c

28.(10 pts)More of the same.

a � b2 � a � b 2

a2 � b2 � a � b 2a � b � c a �&� b � c a � b � c a �� b � c

29.(10 pts)In mathematics, lower and upper case letters mean

different things. The letter a is not the same as theletter A. Keep that in mind when answering the ques-tions below as in the preceding question.a aa Aa � A A � a

30.(10 pts)Much of this course will center around the manip-

ulation of algebraic expressions, often with the goalof solving an equation. This exercise is the first stepin this direction. Again, indicated with T or F if thetwo expressions are equivalent.a �&� b � c a � b � a � c1 �� a � b 1 � a � 1 � b1 � a � a 1 �� a � a

31.(10 pts)The reason why Mathematics is required for so

many subjects is that it can be used to solve problemsoutside of mathematics, the dreaded word problems.There will be many word problems in this class, usu-ally leading to a mathematical problem of the kind weare discussing at the time. Students don’t like wordproblems because they involve the extra layer of con-verting the word problem to a math problem. Butkeep in mind that math classes are the only kind ofclasses you take where some problems are not wordproblems!This first word problem of this course can be solvedby deriving and solving an equation, but it can also besolved essentially by guessing and modifying the an-swer until it fits, without any algebraic manipulation.We will revisit it in the future in a more complicatedsetting.You buy a pot and its lid for a total of $11. The salesperson tells you that the pot by itself costs $10 morethan the lid. The price of the pot is $ and theprice of the lid is $ .


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Tom Robbins.


due 9/9/02 at 11:59 PM.This second home work set will be more typical

than the first in that it deals with the mathematics ofthis class rather than the mechanics of WeBWorK.

To obtain the most benefit from the homeworks Irecommend you do the following:� As soon as the homework opens print a hard

copy. Find a chunk of uncluttered time anda quiet place away from the computer and goover the problems. Figure out their answersand note them down. If there is a problem youcan’t solve, read in your textbook or the rele-vant web pages of this class, talk with friends,obtain help from tutors or myself, and don’tgive up until you figured out how to solve theproblem. Do all this well before the deadlineof the assignment.� When you are ready return to the computerand enter your answers. If WeBWorK rejectsthem you may be able to figure out right thenand there what went wrong, and reenter theanswers. (You will always have an unlimitednumber of attempts.) If there is nothing ob-viously wrong with your answer set it aside,deal with other questions, and then return tothe above described study mode and figureout what’s going on. Then return to the com-puter.

This set is mostly on fractions . Of course you arefamiliar with fractions, and you might think they areobsolete since nowadays we have calculators. How-ever, there are two reasons for including this set.Most importantly, the rules that govern fractionsalso apply to algebraic expressions and handlingthose becomes much easier if you don’t have to worryabout your fluency in fraction rules. Secondarily, it’spart of being part of mathematically literate to have

some sense for numbers, and working with fractionsreinforces that number sense.

Some modern Calculators let you do fractionalarithmetic. However, I recommend you do theseproblems without a calculator to get your brain cellsused to thinking ”rules for fractions” later when youhandle algebraic expressions.

Whenever you encounter words or phrases youdon’t understand, such as maybe algebraic expression, don’t just skip over them but figure out what theymean. This will hold you back momentarily, butit will make you much more efficient overall, andactually save you time in the long run. There is aglossary for this course that has links to relevantweb pages with definitions of the words involved.In most cases any standard dictionary will also tellyou what the words mean. A large part of learningmathematics is learning a language.

There also some basic percent problems in thisset. Many extremely instructive word problems in-volve percent.

Some of the problems in this sets have Hints. Af-ter you submit your first answer a button ”show hint”will appear. If you click on it the system will producea hint after you submit your next answer.

Procrastination is hazardous!Peter Alfeld, JWB 127, 581-6842.1.(10 pts)

The expression2

11� 11

10can be written as a fraction

ab

where the integers a and b have no factor in common,and b is positive.Enter a= and b=

2.(10 pts)The expression

87� 3


ab

1



811 � 9


ab



129 � 7


ab



67� 7


ab



125� 3


ab



69� 3

6

can be written as a fractionab



63� 9


ab



63 � 9

2127� 4


ab



125 � 11

1069� 4


ab


11.(10 pts)The expression 1

2� 2

3� 3

4� 4

5� 5

6� 6

7 is a fractionab where b is positive, and a and b have no commonfactors.Enter a= and b=

12.(10 pts) If you are unfamiliar with the terminol-ogy in this and the next problem learn about it here.The greatest common factor (GCF) of 14 and 21 isand their least common multiple (LCM) is TheGCF of 12 and 42 is and their least common

2

multiple (LCM) is The GCF of 3 and 5 isand their least common multiple (LCM) is

13.(10 pts)The least common multiple of 1 � 2 � 3 � 4 is .The least common multiple of 1 � 2 � 3 � 4 � 5 is .The least common multiple of 1 � 2 � 3 � 4 � 5 � 6 is .The least common multiple of 1 � 2 � 3 � 4 � 5 � 6 � 7 is.

14.(10 pts)Let x

y be a fraction (reduced to lowest terms) that sat-isfies

12� x

y � 56�

Then x= and y=If this problem gives you pause check here for a gen-eral principle of problem solving

15.(10 pts)Let x

y be a fraction (reduced to lowest terms) that sat-isfies

37� x

y � 65�

Then x= and y=16.(10 pts)

The expression 12� 2

3� 3

4� 4

5� 5

6� 6

7 is a fractionab where b is positive, and a and b have no commonfactors.Enter a= and b=

17.(10 pts) List all the factors of 60 in increas-ing sequence. (If you enter a correct factor in thewrong place of the sequence WeBWork will considerit wrong.)

, , , , , , , , ,, , .

18.(10 pts) List all the factors of 105 in increas-ing sequence. (If you enter a correct factor in thewrong place of the sequence WeBWork will considerit wrong.)

, , , , , , , .19.(10 pts) The proper divisors of a natural num-

ber n are those factors of n that are less than n. Forexample, the proper divisors of 10 are 1, 2, and 5. Ifthe sum of the proper divisors of n is larger than nthen n is said to be abundant. If the sum is less thann then n is deficient. If the sum equals n, then n isperfect.

For example, the proper divisors of 10 add to1+2+5=8 and so 10 is deficient. The proper divisorsof 12 add to 1+2+3+4+6 = 16, and so 12 is abundant.The proper divisors of 6 add to 1+2+3 = 6, and so 6is perfect.Perfect numbers have been studied for more than2,000 years. It is still unknown how many perfectnumbers exist. At present only 39 are known and thelargest of them has 18,669,112 digits.

Find the smallest perfect number greater than 6 andenter it here .

20.(10 pts) The last few problems on this set dealwith simple percentage problems.You make $75000 a year, and you get a 10% raise.Your new salary is $ .(Salaries in these problems are randomly chosen andrange from $20,000 to $100,000.)The next year you get a 10% cut. Your new salary is$ . That is % less than your initial salary.

21.(10 pts) After a 5% raise your new salary is$21000. Before the raise your salary was $ .

22.(10 pts) This is a true story! In 2002 theMath Department obtained two new new buildings,the Leroy Cowles Bounding and the T Benny Rush-ing Mathematics Student Center. We had to evalu-ate furniture bids from various manufacturers that of-fered various discounts. One manufacturer offereda ”30+30” percent discount. This means they cut 30percent off the list price and then took another 30 per-cent off the discounted price. Other manufacturersoffered various single discounts. All other things be-ing equal you prefer the single discount if it is greaterthan percent.

23.(10 pts)You and your coworker together make $16 per hour.You know your coworker earns 10 percent more thanyou do. Your hourly wage is $ .After taking Math 1010 your hourly wage is raised to$12. This is a raise of %.After returning to work you can’t help mentioning ca-sually to your coworker than now you make %more than he does.

He responds wistfully that this is as it should besince now you can figure problems like the ones onthis assignment!

3

24.(10 pts) Suppose you obtain 100 percent crediton all WeBWorK assignments in this class. Then theminimum average percentage on the exams (includ-ing the final exam) that will still get you an A in thisclass is percent.Your answer should be an integer between 0 and 100.

Compute the percentage to 1 or 2 digits beyond thedecimal point, and then round it to the smallest in-teger that’s no less than your computed percentage.The information you need to answer this question isgiven in your Course Information.


UR

4


Tom Robbins.


due 9/16/02 at 11:59 PM.

This homework set covers the following topics:� Problem 1: an easy warm up problem, but re-member the rules of arithmetic precedence.� Problems 2–11: The real number line , in-cluding the concepts of greater, less, distance,absolute value, and intervals.� Problems 12-19: Language. Relevant infor-mation is contained in the web pages con-nected with this class, consult the Glossary.Essentially all of the terms used in this classare also defined in any standard dictionary ofthe English language.� Problems 20-25: The structure of the num-ber system, as explained here. We start withthe natural numbers with which we can al-ways add and multiply. We extend the nat-ural numbers to the integers, with which wecan always subtract, in addition to adding ormultiplying. Finally, we extend the integersto the rational numbers with which we cancarry out all four basic operations. However,it turns out that there are numbers that aren’trational, in particular the numbers

�2 and π.

You can see a beautiful and ancient proofthat the square root of 2 is irrational, andone can show similarly that the square rootsof all prime numbers are irrational. One ofthe hardest things to appreciate in this con-text is the fact that the various sets of num-bers are nested. So all natural numbers are in-tegers, but some integers aren’t natural num-bers. Similarly, all integers are rational num-bers, but not all rational numbers are integers.Finally, all rational numbers are real num-bers, but some real numbers are irrational. Infact, at this stage we know of no numbers that

aren’t real, but that will change later in thesemester.� Problems 26-34: Algebraic Expressions. Youcan use them to express formulas, and youcan manipulate them to accomplish amaz-ing feats. There is a brief discussion here.The key to manipulating algebraic expres-sions is the fact that they obey the samelaws as numbers which is the reason, for ex-ample, that you were asked to practice frac-tions in the preceding home work. There willbe much more on manipulating algebraic ex-pressions throughout this course. For some ofthese problems you may have to find out for-mulas, like for the volume of the sphere. Youshould remember these formulas from highschool, but you can also find them in the text-book, a dictionary, or an encyclopedia.� Problem 35: one of those dreaded word prob-lems that are the reason the system makes youlearn math. At this stage the word problemscan be solved by trial and error, but one maintask in this class is to learn systematic ways ofsolving certain types of equations, and applythese techniques to word problems.

As usual, right after this set opens you should printa hard copy and don’t delay working on the set since

Procrastination is hazardous!Peter Alfeld, JWB 127 or 236, 581-6842.

1.(10 pts) As a warm up exercise evaluate3 � � 5 � 6 � 9 � 2 � 6 =Remember that a missing operator means multipli-cation, and multiplication and division come beforeaddition and subtraction unless otherwise indicatedby parentheses. Parentheses can always be includedeven if they are not needed.

2.(10 pts) Evaluate the expression � 80 � =3.(10 pts) Evaluate the expression � � 166 � =4.(10 pts) Evaluate the expression � 129 � 318 �=

.

5.(10 pts) Evaluate the expression �'� 180 � 290 � �� 155 � 307 �(� =1

6.(10 pts) Evaluate the expression ) 136 � 299 )) � 29 ) =. Give you answer in decimal notation (like 1.2345)correct to four decimal places or give your answer asa fraction.

7.(10 pts) The distance between 499 and 165 is.

8.(10 pts) The distance between 279 and 461 is.

9.(10 pts) Enter a T or an F in each answer spacebelow to indicate whether the corresponding state-ment is true or false.You must get all of the answers correct to receivecredit.

1. 10 � 1 * 102. π + 3 � 14163. � 1 � � 104. � 6 � � 65. � 1 * � 1

10.(10 pts)Match the verbal statements given below with the

letters labeling their equivalent inequalities.

1. x is less than 72. The distance from x to 7 is more than 33. The distance from x to 7 is less than or equal

to 34. x is less than or equal to 75. x is greater than 7

A. x * 7B. � x � 7 �,� 3C. x � 7D. 7 � xE. � x � 7 �,* 3

11.(10 pts)Match the intervals with the given inequalities.

1. � 3 � 5 -2. . 3 � 5 3. . 3 � 5 -4. � 3 � 5 A. 3 � x � 5B. 3 * x * 5C. 3 * x � 5D. 3 � x * 5

12.(10 pts) Indicate whether the following state-ments are True (T) or False (F). You must get all an-swers correct in order to receive credit.

1. 2 + 22. 10000 � � 10000003. 2 � 24. 2 * 25. � 2 + � 2 � 5 + � 36. � 4 � 17. � 2 * � 2 � 5 * � 3

13.(10 pts) Indicate whether the following state-ments are True (T) or False (F). You must get all an-swers correct in order to receive credit.

1. 5 � 52. 5 * 53. 5 /� 54. 5 � 55. 5 + 56. 5 � 5

14.(10 pts) Enter the appropriate symbol � , � , or� below.1. 3 � 12. � 2 � 13. x x � 14. x2 � x � 25. 3 � x � 1 3x � 36. � 1 � x �

15.(10 pts) Match the statements defined belowwith the letters labeling particular numbers. Use allthe letters. Of course a natural number is also a ratio-nal number, for example. However, there is only onecorrect matching that uses all five letters A throughE.

1. x is an irrational number2. x is a natural number3. x is an integer4. x is a rational number5. x is neither positive nor negativeA. x � 12B. x � 0C. x � πD. x � 17

12E. x �� 17

16.(10 pts) The next three questions reinforce thevocabulary for the four basic arithmetic operations.

2

Don’t used words like ”minussing” or ”timesing”,they are juvenile. Use the proper terminology em-ployed in this class.

Match the verbs below with the letters labeling par-ticular nouns.

1. multiply2. subtract3. divide4. addA. quotientB. sumC. differenceD. product

17.(10 pts) Match the verbs below with the letterslabeling particular symbols.

1. add2. divide3. multiply4. subtractA. �B. �C. �D. �

18.(10 pts) Match the nouns below with the letterslabeling particular symbols.

1. quotient2. difference3. sum4. productA. �B. �C. �D. �

19.(10 pts) Match the phrases given below with theletters labeling the algebraic expression.You must get all of the answers correct to receivecredit.

1. The sum of x and 2, all squared2. The difference of x and x2 divided by the sum

of x and x2

3. The quotient of x and the sum of x and 84. The sum of x and 85. The product of the sum of x and 2 and the sum

of x2 and 2

A. � x � 2 2B. x

x � 8C. x � 8D. � x � 2 � x2 � 2 E. x � x2

x � x2

20.(10 pts) Match the phrases given below with theletters labeling the algebraic expression. You mayhave to simplify your expression to recognize it asthe correct one.You must get all of the answers correct to receivecredit.

1. The square of x � 32. The difference of x and x � 33. The quotient of x and x � 34. The product of x and x � 35. The sum of x and x � 3A. 2x � 3B. x

x � 3C. x2 � 3xD. � 3E. x2 � 6x � 9

21.(10 pts) Indicate whether the following state-ments are True (T) or False (F).

1.�

3 is a rational number2.�

49 is a rational number3. 2 is a real number4. π is a real number5. 0 is a natural number6. -17 is an integer7. 3

2 is an integer


1. The quotient of two natural numbers is al-ways a natural number.

2. The ratio of two natural numbers is alwayspositive

3. The sum of two natural numbers is always anatural number.

4. The difference of two natural numbers is al-ways a natural number.

5. The product of two natural numbers is alwaysa natural number.

6. The difference of two natural numbers is al-ways an integer.

3

7. The quotient of two natural numbers is al-ways a rational number


1. The quotient of two integers is always an in-teger (provided the denominator is non-zero).

2. The difference of two integers is always aninteger.

3. The sum of two integers is always an integer.4. The ratio of two integers is always positive5. The product of two integers is always an inte-

ger.6. The difference of two integers is always a nat-

ural number.7. The quotient of two integers is always a ra-

tional number (provided the denominator isnon-zero).


1. The quotient of two rational numbers is al-ways a real number (provided the denomina-tor is non-zero).

2. The difference of two rational numbers is al-ways a natural number.

3. The product of two rational numbers is al-ways a rational number.

4. The difference of two rational numbers is al-ways a rational number.

5. The quotient of two rational numbers is al-ways a rational number (provided the denom-inator is non-zero).

6. The sum of two rational numbers is always arational number.

7. The ratio of two rational numbers is alwayspositive


1. The quotient of two real numbers is alwaysa real number (provided the denominator isnon-zero).

2. The sum of two real numbers is always a realnumber.

3. The difference of two real numbers is alwaysa real number.

4. The product of two real numbers is always areal number.

5. The quotient of two real numbers is always arational number (provided the denominator isnon-zero).

6. The ratio of two real numbers is never zero.7. The difference of two real numbers is always

an irrational number.

26.(10 pts) The essence of Algebra is the manipu-lation of algebraic expressions. One application of al-gebraic expressions is the statement of formulas thatdescribe general facts. In the next few questions youare asked to enter some formulas which should be orbecome a permanent part of your Math background.

Enter here an algebraic expression that givesthe area of a rectangle with a length of a and a widthof b. (Use an asterisk to denote multiplication.)

27.(10 pts) Enter here an algebraic expressionthat gives the area of a triangle with a base of lengthof b and a height h.

28.(10 pts) Enter here an algebraic ex-pression that gives the area of a circle with a radius r.You may write ’pi’ to denote the symbol π, and youcan use the symbolˆor a double asterisk ** to denoteexponentiation. (Or you can express r2 as r*r.)Enter here an algebraic expression that gives thearea of a circle with a diameter d.

29.(10 pts) Enter here an algebraic expressionthat gives the area of a sphere with a radius r.Enter here an algebraic expression that gives thevolume of a sphere with radius r.

30.(10 pts) The basic idea of manipulating alge-braic expressions is that they obey the same laws asarithmetic expressions. The following are some sim-ple exercises along those lines. They ask you to enternumerical values for the variables A, B, C ...

The expression 2 � 2 � 3x equals Ax � Bwhere A equals:and B equals:[NOTE: Your answers cannot be algebraic expres-sions.]

31.(10 pts) The expression � 6t � 2 � 6t � 3 � 7t � 3equals At2 � Bt � Cwhere A equals:and B equals:

4

and C equals:

32.(10 pts) The expression � 5x � 2 2equals Ax2 �Bx � Cwhere A equals:and B equals:and C equals:

33.(10 pts) The expression � x � 7 � x2 � 6x �6 equals Ax3 � Bx2 � Cx � Dwhere A equals:

and B equals:and C equals:and D equals:

34.(10 pts) The expression 4 � 6x2 � 7x � 7 �5 � 5x2 � 3x � 7 equals

x2 � x �35.(10 pts) Suppose you go to an event where chil-

dren’s tickets cost $5 and adults tickets cost $10. Youbuy 6 tickets for $40. There are adults andchildren in your party.


UR

5


Tom Robbins.


due 9/23/02 at 11:59 PM.

All of the problems in this set involve the solutionof linear equations. Study this page before com-mencing work on this assignment.

There is a strong flow in this set from simpleproblems to more complicated ones. Many prob-lems can be converted into a simpler kind that getssolved in the problem before. This illustrates ageneral mathematical principle . Over and over inmathematics you reduce your problems to one youhave solved before. Here is a joke that illustrates theconcept. (Being a mathematician I feel I can makefun of mathematicians, but not of other professions.).

Seriously, I recommend that you work throughthese problems in the given order and make sure youunderstand what happened in each one before goingon to the next.

Quite a few of these problems areword problems.

One aspect of some of these problems that may benew to you is that the answer is not some specificnumber but an algebraic expression involving vari-ables. You can think of your answer as a formula.

Remember that the answers, and in most cases de-tailed solutions, of the past homework problems areavailable. They may be useful in the solution of someof the problems on this set. Go back to the problem,click on ”show solution” and ”show answers”, andthen click on ”submit answers”.

Most of the questions in this set contain hints thatyou can see after you submit your first answer. Mostof them also have solutions that you can see after theset closes.

Problem 32 differs from all the others. It is de-signed to help you prepare for Exam 1.

1.(10 pts)The solution of the equation 7x � 8

is x � .You may enter your answer as a decimal number oras a fraction. I recommend that in problems like thisyou use fractions rather than decimal approximations.You don’t have to figure out your approximation, andyou don’t have to worry about just how accuratelyyou should approximate the answer.

2.(10 pts)The solution of the equation 5x � 2 � 5is x � .

3.(10 pts)The solution of the equation 7x � 3 � 6x � 9is x � .(You may enter your answer as a decimal number oras a fraction.)

4.(10 pts)The solution of the equation 8y � 13 � 3y � 12is y � .(You may enter your answer as a decimal number oras a fraction.)

5.(10 pts)You are probably used to solving problems where thecoefficients are specific numbers. However, in manyproblems the coefficients are variables themselves,and the answer depends on those variables. As yougo on in mathematics, the role of specific numberswill keep decreasing, and the role of general coeffi-cients (or parameters) will increase. The next coupleof problems are our first foray into this new area.The solution of the equation ax � b � cis x � .(Your answer will of course be in terms of a, b, andc. You may assume that a is non-zero.)

6.(10 pts)The solution of the equation ax � b � cx � dis x � .(Your answer will of course be in terms of a, b, c, andd. You may assume that a does not equal c.)

7.(10 pts)The solution of the equation

x2 � 4x � 13 � x2 � 3x � 19

is x � .8.(10 pts)

1

The solution of the equation

t2 � 10t � 15 � t2 � 7t � 17

is t � .(You may enter your answer as a decimal number oras a fraction.)

9.(10 pts)The solution of the equation� x � 4 � x � 5 � � x � 3 �� x � 3 is x � .(You may enter your answer as a decimal number oras a fraction.)

10.(10 pts)The solution of the equation� z � 5 � z � 8 � � z � 9 � z � 9 is z � .(You may enter your answer as a decimal number oras a fraction.)


1x � 10 � 1

2x � 3is x � .


1x � 6 � 2

x � 9is x � .


1x2 � 4x � 6 � 1

x2 � 5x � 6is x � .


1z2 � 8z � 2 � 1

z2 � 3z � 6is z � .


xx � 2 � x � 7

x � 3is x � .

(You may enter your answer as a decimal number oras a fraction.)


x � 1x � 2 � x � 1

x � 7is x � .(You may enter your answer as a decimal number oras a fraction.)

17.(10 pts) Indicate with T (true) if the equationsbelow are linear, and F (false) if they are not.

1. x2 � 12. 3x � 4 � 173. x2 � x � 44. 3x � 4 � 5x � 4

18.(10 pts) Indicate with T (true) if the equationsbelow are equivalent to a linear equation and with anF (False) if not. For example, the equation

x2 � 3x � 4 � x2 � 5

turns into the linear equation

3x � 4 � 5

after subtracting x2 on both sides.Actually, the issue of whether a given equation isequivalent to a linear equation is quite subtle, but forthe problems given here the answer will be (hope-fully) obvious. To help you along, in this problemWeBWorK will indicate for each answer separatelywhether it’s right or wrong and so you can get creditby just trying T’s and F’s. Of course you should thinkabout the questions carefully and figure out why theanswer is T or F. This will enable you to approach thenext question more effectively.

1. x2 � 3x � 4 � 17x � 5 � x2

2. 1x � 1 � 1

2x � 13. x2 � 4x � 2 � 6x � 14 � x2

4. x6 � 2

3 � 56

19.(10 pts) Indicate by T or F whether the equa-tions below are equivalent to a linear equation. Thistime you need to get all answers correctly before re-ceiving credit.

1. x2 � 5x � 4 � x2 � 6x � 112. 1

x � 3 � 12x � 3

3. x2 � 1 � x � x2

2

4. 3x � 4 � 5x � 4

20.(10 pts) For the following equations, enter Uif the equation has a unique solution, N if it has nosolution, and I if it has infinitely many solutions.You need to get all answers correct before obtainingcredit.

1.4x � 5 � 42.4x � 4 � 4x � 5

3. 14x � 5 � 1

5x � 44. 1

4x � 5 � 24x � 5

5. � x � 4 �� x � 5 � x2 � 9x � 206.2 � 4x � 5 � 8x � 10

21.(10 pts) The remaining problems in this set areword problems . You may want to look for similarproblems in the textbook for this class, if you have it,or any other textbook on Intermediate Algebra.You are hiking along the California coast and wonderabout the height of a particular Giant Redwood tree.You are 5 feet and nine inches tall and your shadow is5 feet long. The shadow of the tree is 180 feet long.How tall is the tree? Enter its height here: feet.

22.(10 pts)You can paint a certain room in 7 hours. Your brothercan do it in 2 hours. How long does it take the two ofyou working together?

hours.23.(10 pts) How much 90 percent vinegar do you

have to add to a gallon of 5 percent vinegar to get 20percent vinegar?

gallons.24.(10 pts) Enter here an algebraic expression

that gives the area of a circle with a circumference c.25.(10 pts) Suppose your average percentage on

these WeBWorK homeworks is p percent. You wishto obtain a grade in this class that requires you to ob-tain at least q percent overall. Enter here theminimum average percentage your need to obtain onthe exams. Your answer should of course depend onp and q.

26.(10 pts)This problem is just like the pot and lid problem onhome work 1.You buy a house including the land it sits on for$176000. The real estate agent tells you that the landcosts $8000 more than the house.

The price of the house is $ and the price of theland is $ .

27.(10 pts)You figure that you will buy property many times andyou endeavor to work out a formula that will tell youthe price of the land and the price of the buildings anytime.So suppose the price for the land and buildings com-bined is S (for ”sum”) and the land costs D (for ”dif-ference”) more than the buildings.

Then the price of the buildings is dollarsand the price of the land is dollars.Remember that in mathematics upper and lower

case letters are distinct. So make sure that in yourformulas you use upper case letters S and D, ratherthan lower case letters!

28.(10 pts)Suppose the next time you buy a house including theland it sits on for 176000 Dollars. The real estateagent tells you that the land costs 17000 more thanthe house.

You apply your formula derived in the previousproblem. To do so you set S � dollars and D �

dollars.and you find out that the price of the house isdollars and the price of the land is dollars.

29.(10 pts)The next time you buy a house including the land for122000 Dollars. This time the real estate agent tellsyou that the land costs 6000 Dollars less than thehouse.You realize that you can still apply your formula. Youset S � dollars and D � dollars.and you find out that the price of the house isdollars and the price of the land is dollars.You consider your recent purchase a great deal, yourealize that you just developed a mathematical theoryof buying combinations and figuring individual pricesin terms of sums and differences, you recognize thatMath 1010 is a useful class, and you sit back in yournew house, feeling well pleased with yourself, andmarveling at the unreasonable power of mathematics!

30.(10 pts)The last two problems are examples of ”simple

Hindu Algebra”, quoted on page 528 of ”The Story of3

Civilization”, v.1., by Will Durant, Simon and Schus-ter, 1935. The problems are approximately 1,800years old. (Mental pursuits by women were discour-aged at that time. You ponder the significance offictiously directing the questions at women. Durantdoes not comment on this issue.)Out of a swarm of bees one fifth part settled on aKadamba blossom; one third on a Silihindra flower;three times the difference of those numbers flew tothe bloom of a Kutaja. One bee, which remained,hovered about in the air. Tell me, charming woman,the number of bees .

31.(10 pts) Here is the other problem from ”TheStory of Civilization”:Eight rubies, ten emeralds, and a hundred pearls,which are in thy ear-ring, my beloved, were pur-chased by me for thee at an equal amount; and thesum of the prices of the three sort of gems was threeless than half a hundred; tell me the price of each,auspicious woman.Enter the price of one pearl , the price of oneemerald , and the the price of one ruby .

32.(1 pt) The purpose of this last and lengthy prob-lem is to help you prepare for the first exam. Ifyou work through this problem, follow the links, andmake sure you understand what you read you will dowell on the exam. There’s no reason you shouldn’tget 100 percent! Let me know if you have any ques-tions!

There will be ten questions, all questions haveequal weight, and you’ll be able to use only pen orpencil and the exam itself, i.e.,no notes, books, orcalculators! Work through the list below and makesure you can solve correctly and easily problems likethose listed.

Before the exam, read the online guidelines for exams.During the exam, follow those guide-lines!

Here is a list describing the problems on the exam:� Fractions. You need to be able to add, multi-ply, subtract, and divide fractions. Moreover,you need to be able to recognize common fac-tors in numerator and denominator, and can-cel them. It does not hurt to practice, so herewe go:27� 4

21 � � .27 � 4

21 � � .

27� 4

21 � � .27� 4

21 � � .� The Number System. You need to under-stand the number system and how it is builtstarting with the natural numbers, and thenconstructing integers, rational numbers, and,finally, real numbers. Understand what thosewords mean, and that all natural numbers areintegers, all integers are rational numbers,and all rational numbers are real numbers.Some real numbers aren’t rational, some ra-tional numbers aren’t integer, and some inte-gers aren’t natural numbers. For some rea-son this tends to be very confusing, but it’sjust like saying that all women are people, butsome people aren’t women. (The same goesfor men, of course, and then there are kidstoo.) Never mind the people, but be preparedto give examples of these various types ofnumbers.� Language. Understand the language re-lated to the four basic arithmetic operations.Don’t use embarrassingly juvenile words like“timesing” and “minussing”. Use the properterms multiplying and subtracting. Under-stand the words add, multiply, subtract, di-vide, sum, difference, product, quotient, fac-tor, multiple, divisor, dividend. Use theGlossary and have someone quiz you if youare not sure.� The Distributive Law. Understand how toapply the Distributive Law to simplify al-gebraic expressions. Examples are given inproblems 30–34 on set 3. (I sometimes thinkthat instead of “Intermediate Algebra” thiscourse should be called “A First and GentleIntroduction to Applications of the Distribu-tive Law.”)� Precedence. Understand the conventions ofarithmetic precedence. There are a largenumber of problems like this on set 1. Multi-plication and division come before additionand subtraction. If there are several oper-ations of the same level of precedence wework from left to right. However, anythingin parentheses is evaluated first. Addition

4

and subtraction have the same level of prece-dence, and so do multiplication and division.You can also use superfluous parentheses tomake your meaning clear if you are not sure.� Linear Equations. To solve an equationmeans to figure out the values of the variablethat make the equation true. There’s just oneprinciple involved in equation solving: youfigure out what bothers you and get rid ofit by doing the same thing on both sides ofthe equation. Knowing what bothers you ishalf the battle. Then you undo multiplica-tion by division, division by multiplication,addition by subtraction, subtraction by addi-tion, fractions by multiplication or formingreciprocals. (The list goes on, for exampleyou undo square roots by squares, squares bysquare roots, absolute values by case distinc-tions..., but we aren’t there yet.) Eventuallyyou come up with a statement that the vari-able equals some specific value. There aremany problems like that on this set. How-ever, for good measure try your hand also onthis equation:If 3x � 5 � 1 then x � (Enter youranswer as a fraction.)� The coefficients of a linear equation aren’t al-ways nice, so here is another equation:If 2x

7� 5

3 � 1 then x � (Again, enteryour answer as a fraction.)

� Sometimes we need to manipulate an equa-tion a bit before recognizing it as a linearequation. Terms that appear to be nonlinearmay disappear. Problems 7–10 on this set arelike that.� The phrase ”doing the same thing on bothsides” includes operations like forming recip-rocals or multiplying with common denom-inators. Problems 11-16 on this set are likethat. (However, when solving equations wenever do one thing on one side, and some-thing different on the other.)� There is a word problem on the set. It’s aboutbuying a car, and I realize that you may neverhave bought a car. The problem also con-tains some words you may not have seen be-fore, like ”invoice price” (which is the dol-lar amount the dealer is charged by the man-ufacturer) and ”factory incentive” (which isa rebate given by the manufacturer to thedealer). However, the problem can be solvedjust using common sense and understandingthe language of percent.Here is another percent problem: You buy aused car listed as $10,000 at a discount of 10percent. However, you also pay 10 percentsales tax. Thus you end up writing a checkfor $ .


UR

5


Tom Robbins.


due 9/30/02 at 11:59 PM.

This home work set covers the following topics:Solving Inequalities. The basic idea is that you

process inequalities like equations, except that youreverse the inequality when you multiply on bothsides with a negative number.

Absolute Value Equations. Here the basic idea isthat for an absolute value to equal a certain positivenumber there are always two possibilities. You solveabsolute value equations by considering all possiblecases.

A first introduction to the concept of functionswhich is central and crucial in mathematics.

1.(10 pts)Consider the inequality

3x � 7 � 2By subtracting 7 on both sides and dividing by 3

we see that this inequality is equivalent to

x � � 53�

The first few problems in this set are similar. In theabove example you would enter � and � 5

3 as youranswers.In the actual first problem of this set consider the in-equality

2x � 9 � 9Below insert the appropriate symbol � or � and theappropriate number such that the two inequalities areequivalent.x (insert symbol) (insert number)As usual I recommend that you enter non-integer an-swers as fractions rather than decimal expressions.

2.(10 pts)Consider the inequality� 8x � 5 � 8

Below insert the appropriate symbol � or � and theappropriate number such that the two inequalities areequivalent.x (insert symbol) (insert number)


6x � 9 � 13x � 2



7x � 5 � 2x � 6


5.(10 pts)To solve the next few problems you needto understand the definition and properties ofabsolute value.The equation � 4x � 7 � � 7has two solutions. Enter the smaller here andthe larger here

6.(10 pts)The equation � 2x � 9 � � � 5x � 6 �has two solutions. Enter the smaller here andthe larger here

7.(10 pts)The equation � 2x � 3 � � � 7x � 7 �has two solutions. Enter the smaller here andthe larger here

8.(10 pts)The equation � x � 1 � � � x � 2 � � 3 �has two solutions.

Enter the smaller hereand the larger here

1

9.(10 pts)The equation� x � 1 � � � x � 2 � � � x � 3 � � 4 �

has two solutions. Enter the smaller hereand the larger here:10.(10 pts) Three numbers may be the lengths of

the side of a triangle if the largest side is shorter thanthe sum of the two smaller. Indicate as true (T) offalse (F), whether the following triples of numbersmay be the length of the sides of a triangle.

You need to get all answers correct before obtain-ing credit.

3 � 4 � 51 � 2 � 43 � 3 � 5

17 � 17 � 172 � 17 � 18

11.(10 pts) For the next few problems you need tounderstand what it means to evaluate a function. Yousimply replace the value of the variable with the num-ber at which you evaluate the function. For example,the answer to the first question below is 56 since

56 � 9 � 6 � 2 �Let the function f be defined by

f � x � 9x � 2 �Then f � 6 � and f � 7 �

12.(10 pts)Let the function f be defined by

f � x �� 4x � 6 �Then f � � 4 � and f � � 3 �


f � x � 7x � 7 �Then f � 3 � f � 5 � and f � 3 � f � 5 �


f � x � 2x � 5 �Then f � 7 � f � 5 � and f � 7 � f � 5 �

15.(10 pts)

Let the function f be defined by

f � x � 5x � 8 �Then f � 6 � f � 8 � and f � 6 � 8 �


f � x � 3x � 2 �Then f � 3 � f � 2 � and f � 3 � 2 �


f � x � 8x � 6 �Then f � x � 1 � and f � x � 1 �


f � x � 5x � 7 �Then f � f � 9 � and f � f � 10 � �

19.(10 pts) The next few problems are exercises inidentifying the (natural) domain of a function. Theconcept of a function is very general but for our pur-poses the inputs and outputs of a function are realnumbers. The domain is the set of real numbers atwhich the function can be evaluated, and the rangeis the set of all possible outputs. To determine thedomain ask yourself at what points the function canNOT be evaluated. Usually this is because of an un-defined operation, which practically speaking meansdividing by zero, or extracting the square root of anegative number.Let the function f be defined by

f � x � 7x � 93x � 9

�The the domain of f contains all real numbers x ex-cept x �


f � x � � 6x � 5 �Then x is in the domain of f provided x +


f � x � � � 3x � 2 �Then x is in the domain of f provided x *

2

22.(10 pts) Let the function f be defined by f � x �10

1 � x2 � Indicate whether the following statements areTrue (T) or False (F). You must get all answers cor-rect in order to receive credit.

1. 1 is in the domain of f2. f � x is never zero.3. All negative real numbers are in the domain

of f4. All positive real numbers are in the domain of

f5. f � x is never positive.6. 0 is in the domain of f7. f � x is never negative.

23.(10 pts) Some equations involving x and y de-fine y as a function of x, and others do not. For ex-ample, if x � y � 1 we can solve for y and obtainy � 1 � x and we can then think of y � f � x � 1 � x.On the other hand, if we have the equation x � y2 then

y is not a fucntion of x since for a given non-negativevalue of x the value of y could equal the positive orthe negative square root of x. Indicate by True (T) orFalse (F). whether the following equations define y asa function of x. You may assume that 0 * x * 1 �

1. � y � � x � 02. y2 � x2 � 13. y � � x � � 04. x � y � 15. 5x � 4y � 8 � 06. y2 � x � 17. y � x2 � 1

24.(10 pts) You are going to drive 480 miles todayat an average speed of 60 miles per hours. Thus youare going to drive for a total of hours. Thediameter of the wheels on your car (including tires)is 25 inches. Thus each wheel is going to turn a totalof times today.


UR

3


Tom Robbins.


due 10/7/02 at 11:59 PM.

This homework set deals with the Cartesian (orRectangular) Coordinate System and the graph ofequations and functions, particularly with the graphsof linear equations which are straight lines. Thepower of the approach stems from the fact that Carte-sian Coordinates provide a close link between ge-ometry and algebra, thereby greatly amplifying yourproblem solving ability.

The material in this home work set is covered in thefollowing web pages, some of which are interactive.� A review and proof of the Pythagorean Theorem.� An introduction to the Cartesian Coordinate System.� An interactive page that lets you plot a point

given the coordinates, or find the coordinatesgiven the point.� An introduction to graphs and equations ofstraight lines.� Another interactive page that lets you drawa line given some information about it, orshow you a line and lets you figure out itsproperties.� An introduction to the concept of functions.We covered this subject in an earlier home-work.� An introduction to the concept of thegraph of an equation or function, illus-trated in the special case of parabolas.� Yet another interactive page that lets youexplore the interplay between simple alge-braic changes and their effects on the graph.� If you are curious, here is an introduction toconic sections (or conics, for short) whichexpands the earlier discussion of parabolas.

1.(10 pts) I recommend that instead of decimalnumbers you enter an expression using sqrt() to in-dicate a square root. For example instead of 1.4142you would enter sqrt(2).

The distance between the points � 5 � 1 and � 1 � 9 is.

The distance between the points � 4 � 5 and � 1 � 1 is.

The distance between the points � � 5 � 4 and � 6 � 5 is .

The distance between the points � � 9 � � 8 and� � 7 � � 1 is .

2.(10 pts) Each of the following phrases describesa point in the Cartesian (rectangular) coordinate sys-tem. Enter the coordinates x and y of the point.The point is located 5 units to the left of the y axisand 2 units above the x axis.x= and y=The point is located 10 units to the right of the y axisand 4 units below the x axis.x= and y=The coordinates of the point have equal absolutevalue, the point is in the second quadrant, and thedistance of the point from the origin is 2.x= and y=Hint: use ”sqrt(z)” to denote the square root of a num-ber z.The point is the origin.x= and y=The point is on the positive x axis 10 units from theorigin.x= and y=

3.(10 pts) The slope of the line through the points� 3 � 1 and � 5 � 8 is .The slope of the line through the points � 9 � 7 and� 10 � 3 is .The slope of the line through the points � � 6 � 5 and� 3 � 7 is .The slope of the line through the points � � 7 � � 8

and � � 3 � � 9 is .

4.(10 pts)The line defined by the equation y � 2x � 3 has thex-interceptand the y-intercept .

1

The line defined by the equation y � 5 � 5x � 2 � 5 hasthe x-interceptand the y-intercept .

5.(10 pts)The equation

5x � 4y � 7 � 0has the x-intercept and the y-intercept .It defines a straight line of slopeThe line defined by the equation

9 � 5x � 5 � 3y � 4 � 1 � 0

has the x-intercept and the y-intercept .Its slope isThe line defined by the equation� 4 � 2x � 7 � 4y � 7 � 6 � 0

has the x-intercept and the y-intercept .Its slope is

6.(10 pts) For the lines defined by the followingequations indicate with a ”V” if they are vertical, an”H” if they are horizontal, and an ”S” (for slanted) ifthey are neither vertical nor horizontal.

3x � 4y � 5 � 04y � 5 � 03x � 5 � 0

x � 1y � 1y � x

7.(10 pts) For the pairs of lines defined by the fol-lowing equations indicate with an ”I” if they are iden-tical, a ”P” if they are distinct but parallel, an ”N” (for”normal”) if they are perpendicular, and a ”G” (for”general”) if they are neither parallel nor perpendic-ular.

3x � 4y � 5 � 0 and 6x � 8y � 10 � 0.3x � 4y � 5 � 0 and 3x � 4y � 7 � 0.3x � 4y � 5 � 0 and 3x � 5y � 7 � 0.� 4x � 3y � 5 � 0 and 3x � 4y � 7 � 0.

8.(10 pts) For the pairs of lines defined by the fol-lowing equations indicate with an ”I” if they are iden-tical, a ”P” if they are distinct but parallel, an ”N” (for”normal”) if they are perpendicular, and a ”G” (for”general”) if they are neither parallel nor perpendic-ular.

3x � 4y � 5 � 0 and y �� 34x � 5

4 .x � � 2 and y � π.

y � x � 1 and 3x � 5y � 7 � 0.y �� 3

4 x and 3x � 4y � 7 � 0.9.(10 pts)

The slope-intercept equation of the line through thepoints � 5 � 6 and � 2 � 9 isy � mx � b

where m � and b � .10.(10 pts)

The slope-intercept equation of the line through thepoints � 3 � 1 � � 6 � 3 and � � 3 � 5 � 3 � 7 isy � mx � b

where m � and b � .11.(10 pts) The next few questions will reinforce

your mastery of the Cartesian Coordinate Systemand of equations of straight lines .

The distance between two points is obtained bythe Pythagorean Theorem. The distance of a pointP from a line L is the shortest distance between thatpoint and a point on the line. Geometrically, you canobtain it by drawing a line through P perpendicularlyto L. It will intersect L in a point Q which is the pointon L closest to P. Once you have Q you simply com-pute the distance h between P and Q. These conceptsare illustrated in this Figure:

We will build slowly to a general formula for the dis-tance of P from L. Let’s start with the line L definedby

y � 12

x � 1 �The slope of L isy � .A line that’s perpendicular to L has the slope .

12.(10 pts) Consider again the line L defined by

y � 12

x � 1 �2

The line that is perpendicular to L and passesthrough the point � 3 � 0 can be written inslope-intercept form asy � x � .

13.(10 pts) The line defined by

y � x2� 1

and the line defined by

y � � 2x � 6

intersect in the point � , .14.(10 pts) The distance of the point � 3 � 0 from the

liney � 1

2x � 1

is .15.(10 pts) The next three problems have you redo

the steps illustrated in the previous problems for dif-ferent lines and different points. You can do themright away, or first do problem 18 and and then applythe formulas that you’ll derive there.The point � 0 � 1 has the distance , from the linedefined by

y � � 2x � 1

16.(10 pts) The point � 1 � 2 has the distancefrom the line defined by

y �� x3� 1 �

17.(10 pts) The point � � 3 � 4 has the distance

from the line defined by

y � � 2x � 2

18.(10 pts) This question is the capstone of the pre-vious few problems. In the last question you had todo quite a few calculations. If you were to computefrequently the distance of a point from a line it wouldbe handy to have a formula into which you plug thecoordinates of the point and the equation of the line.At the end of this problem you will obtain such a for-mula. To help you along you can use WeBWorK tocheck your intermediate answers.Let P be the point � p � q and L the line y � mx � b.It is not necessary, but if you like you may assumethat P lies below the line L. All your answers belowshould be algebraic expressions in terms of m, b, p

and q. The slope of L is . The slope of a lineperpendicular to L is . The line through Pperpendicular to L can be written as y � sx � c wheres is and c is: . That line intersects L in thepoint Q � � u � v , where u is: and v is:

. The distance of P and Q is. The expression you enter here may be quite

messy. However, if it is correct it can be simplifiedinto a very concise and meaningful form. Make sureyou check the solution of this problem when the setcloses.

19.(10 pts) Let F denote a certain temperature indegrees Fahrenheit, and C the same temperature indegrees Celsius. Then you can convert between Fand C by the formula

F � 32 � 95

C �Suppose the temperature is 21 degrees Celsius.Enter here the corresponding temperature indegrees Fahrenheit.

20.(10 pts)Suppose the temperature is 35 degrees Fahrenheit.Enter here the corresponding temperature indegrees Celsius.

21.(10 pts) There is a temperature for which thenumerical values of degrees Fahrenheit and degreesCelsius are equal. Enter that numerical value here

.22.(10 pts)You are working on a new temperature scale that

will unify the earth. After some thought you decideto call it the Tom Robbins-universal-scale. Let Fdenote the temperature in degrees Fahrenheit, and letX denote your new temperature scale. You want it tobe such that if F � 0 then X � 7 and if F � 100 thenX � 111. You also want X to be such that if you plotX against F you obtain a straight line. (This is de-scribed as ”linear interpolation” in many textbooks.)You obtain the formulaX � mF � bwhere m =and b =

23.(10 pts) Match the functions with their graphs.1. F � x � x � 12. F � x � x � 1

3

3. F � x �� x � 14. F � x �� x � 1

A B C D

(Click on image for a larger view )24.(10 pts) Match the functions with their graphs.

1. F � x � 2x � 12. F � x � x � 23. F � x � 2x � 14. F � x � x � 2

A B C D

(Click on image for a larger view )25.(10 pts) Match the equations with their graphs.

1. 2y � 2x � 2 � 02. 2y � 2x � 4 � 03. y � x � 1 � 04. y � x � 1 � 0

A B C D

(Click on image for a larger view )26.(10 pts) Match the functions with their graphs.

1. F � x � x2

2. F � x � � x � 1 23. F � x � x2 � 14. F � x � � x � 1 2

A B C D

(Click on image for a larger view )27.(10 pts) Match the equations with their graphs.

1. x2 � y2 � 12. � x � 1 2 � y2 � 13. x2 � � y � 1 2 � 14. � x � 1 2 � � y � 1 2 � 1

A B C D

(Click on image for a larger view )28.(10 pts) Suppose you are given a function y �

f � x , and you consider shifting and reflecting thegraph of f . Indicate whether the following statementsare true (T) of false (F).

The graph of g � x � f � x � 1 is obtained byshifting the graph of f up one unit.

The graph of g � x � f � x � 1 is obtained byshifting the graph of f down one unit.

The graph of g � x � f � x � 1 is obtained byshifting the graph of f right one unit.

The graph of g � x � f � x � 1 is obtained byshifting the graph of f left one unit.

The graph of g � x � f � � x is obtained byreflecting the graph of f in the y axis.

The graph of g � x �� f � x is obtained byreflecting the graph of f in the x axis.

29.(10 pts) You build a box that is 4 feet long, 5feet wide, and 8 feet high. The distance from the topleft front corner to the top right back corner isfeet.The distance from the bottom left front corner to thetop right back corner is feet.


UR

4


Tom Robbins.


due 10/14/02 at 11:59 PM.The focus of this home work set is on

powers and radicals An application of powersis scientific notation. An application of radicals isscaling. Study all these links to prepare for doingthis home work.

There is a large number of word problems, mostof which are very simple. You may find it useful toconsult these worked word problems .

1.(10 pts) For some of the problems in this setyou may have to evaluate radical expressions. Yourcalculator may be able to do this, or you can enterexpressions like 10

13 into WeBWorK as 10ˆ(1/3) or

10**(1/3). Note the parentheses around the expo-nent! If the exponent is 1

2 you may use sqrt instead.For example, you can say sqrt(7) for 7

12 which is the

same as�

7.Let’s practice entering radicals:Enter here the number 2

12 .

Enter here the number 323 .

Enter here the expression x23 .

Enter here the expression xpq .

2.(10 pts) Evaluate the following powers and en-ter them as integers .32 �23 �

3.(10 pts) Evaluate the following powers and en-ter them as integers .52 �25 �

4.(10 pts) Find a natural number x � suchthat

xx � 27 �5.(10 pts) Find a natural number x � such

thatxx � 1 � 1024 �

6.(10 pts) Note that in general ab does not equala � b.Evaluate the following arithmetic expressions and en-ter them as an integer32 � 3 � 2 = . 62 � 6 � 2 = .

7.(10 pts) Note that in general � a � b 2 doe notequal a2 � b2.Evaluate the following arithmetic expressions and en-ter them as an integer� 3 � 2 2 � � 32 � 22 � .� 9 � 5 2 � � 92 � 52 � .

8.(10 pts) Insert the appropriate symbol � or � .32 23

102 210

990 909

9.(10 pts) Find two natural numbers a and b suchthat b � a and

ab � ba �Then a �and b � .

10.(10 pts) Enter numerical values for the follow-ing powers . I recommend you don’t use a calculator,to make sure you understand the concepts involved.Your answer needs to be a natural number, the sys-tem will not accept an arithmetic expression.9

32 � .

853 � .

2743 � .11.(10 pts) Enter numerical values for the follow-

ing powers .� 52 32 � .� 23 53 � .� 33 43 � .

12.(10 pts) Rewrite the following expressions us-ing just one rational exponent. Enter the numeratorand denominator of the exponent. Cancel any com-mon factors.u2u

32 � u

ab where a � and b � .

z3z13 � z


13.(10 pts) Rewrite the following expressions us-ing just one rational exponent. Enter the numeratorand denominator of the exponent. Cancel any com-mon factors.� u 3

2 27 � u


1

� z 47 7

6 � zab where a � and b � .

14.(10 pts) Rewrite the following expressions us-ing just one rational exponent. Enter the numeratorand denominator of the exponent. Cancel any com-mon factors.� u 3

2 45 � u

ab where a � and b � .� z 4

3 92 � z


15.(10 pts) Rewrite the following expressions us-ing just one rational exponent. Enter the numeratorand denominator of the exponent. Cancel any com-mon factors.u

32 u

27 � u


u32 u

13 � u


u14 u

16 � u



32

u27� u


u32

u13� u


u14

u16� u



23 u

14

u12 u

16� u


u34 u

13

u23 u

56� u



72 u

14

u13 u

1712� u


u154 u

13

u12 u

1912� u


19.(10 pts) The next few problems ask you to solveradical equations .The solution of the equation�

x � 4 � 0

is .

20.(10 pts) The solution of the equation�2x � 1 � 5 � 0

is x � .21.(10 pts) The solution of the equation� 2x � 1 1

3 � 3 � 0

is x � .22.(10 pts) Your banker tells you that your (one

time) investment will double every 12 years. You fig-ure that the effective annual interest rate ispercent. The effective annual interest rate is the profityou obtain at the end of each year, figured as a per-centage. Of course you realize that each year youreceive interest also for the interest earned in the pre-ceding years.

23.(10 pts) Your village depends for its food on anearby fishpond. One day the wind blows a waterlilly seed into the pond, and the lilly begins to grow.It doubles its size every day, and its growth is suchthat it will cover the entire pond (and smother the fishpopulation) in 45 days. You are away on vacation,and the villagers depend on your vigilance and intel-ligence to handle their affairs. Without you presentthey will take action about that lilly only on regularwork days, and only when the lilly covers half thepond or more. The lilly will cover half the ponddays after it first starts growing. The day it reachesthat size happens to be a holiday....

24.(10 pts) In the following three problems enteryour answers with at least four digits.You propose to build a space ship that allows the en-tire population of the United States (280 million) toleave the earth. Your plans allocate an average spaceof 1 � 000 � � 10 � 10 � 10 cubic feet to each person.Your space ship is going to have the shape of a sphere.The radius of your space ship is feet.

25.(10 pts) After due consideration Congress de-cides to support your spaceship plans if you providea cabin with a skylight for each of the 280,000,000passengers. You figure that on average each passen-ger must have 100 square feet on the surface of yourspace ship. The radius of your new and improvedspace ship is feet.

26.(10 pts) You tell Congress that the space shipthey ask for would have a radius that’s times

2

bigger than what you suggested and the volume (andweight and cost) would be as large. Aftersome discussion you decide to build a space ship forjust yourself and leave earth to explore the universe.

27.(10 pts) Complete the following equation. Youranswers will be algebraic expressions.

1a � bi � � i

28.(10 pts) This exercise concerns scientific notation. Fill in the blanks:0 � 0002323 � 2 � 323E .0 � 000000003831 � 3 � 831E .17870000 � 1 � 787E .38030000000 � 3 � 803E .

29.(10 pts) This is like the preceding problem ex-cept you need to get all answers correct to get credit.0 � 003377 � 3 � 377E .0 � 00000000008153 � 8 � 153E .636500000 � 6 � 365E .3795000 � 3 � 795E .

30.(10 pts) Here are some actually occurring num-bers. Write them in scientific notation. Rememberthat there is one digit before the decimal point, and itis non-zero.The land area of Earth is 57,500,000 = Esquare miles.The ocean area of Earth is 139,400,000 = Esquare miles.The length of a light year is 9,461,000,000,000,000 =

E kilometers.This equals approximately 5,880,000,000,000,000 =

E miles.The thickness of the skin of a soap bubble is approx-imately 0.0000001 = E meters. (A meter is alittle more than three feet.)An ounce of Oxygen contains approximately686,000,000,000,000,000,000,000 = Emolecules.

31.(10 pts) Earth weighs about 5.974E21 metrictons. The sun weighs approximately 1.99E27 met-ric tons. The sun weighs times as much as theearth. Enter your answer with at least four digits.

32.(10 pts) You have a 5 foot long ladder, and youlean it against a wall such that the bottom of the lad-der is 3 feet away from the wall. The top of the laddertouches the wall at a height of feet.

Note: The greatest obstacle to progress in mathe-matics is the human inability to distinguish reliablybetween a plus and a minus sign. Make sure youcheck the solution of this problem after the set closes!

33.(10 pts) A 10-foot plank is used to brace a base-ment wall during construction of a home. The plankis nailed to the wall 6 feet above the floor. The slopeof the plank is .

34.(10 pts) It takes you 1 gallon of paint to paintthe inside of a box. The next day you paint a box thathas the same ratios of length to width and length toheight, but it takes you 2 gallons to paint the box. Youconclude that the total inside area of the larger box istwice that of the smaller box. You conclude that thevolume of the larger box equals times that of thesmaller box.

35.(10 pts) You have two boxes of the same shapethat hold flour. The larger box holds twice as muchflour as the smaller one. You conclude that it wouldtake you times as much paint to paint the largerbox than it would take you to paint the smaller box.

36.(10 pts) You have a cube of gold. Your friend(even though she may not be your friend muchlonger) has a cube of gold that is twice as long (andwide and high). Your friend’s cube weights asmuch as yours, and it has times the surface area.You also have a spherical diamond. Your friend hasa spherical diamond of twice the diameter of yours.Your friend’s diamond weights as much asyours, and it has the surface area.

37.(10 pts) Your friend has a cube of gold. Youhave a cube of gold that weighs twice as much. Yourcube is as long as your friends, and it has

times the surface area. You can imaginehow this would work with a spherical diamond, so wewon’t repeat that part here. Here is another question:The ratio of the weight of your cube and the surfacearea of your cube is times as large as thecorresponding ratio of your friend’s cube.

38.(10 pts) The effects of scaling, explored in theprevious two problems for cubes and spheres, alsoapply to odd shaped objects. Just think of them ascomposed of many small cubes and ask what happensto the cubes as you change their lengths.

3

In the novel Gulliver’s travels by Jonathan Swift,Gulliver visits a country, Brobdingnag, in which ev-erything, including people, plants, buildings, etc., is12 times as large as in England. (In that same novelGulliver visits several other strange countries, and ina conversation with the King of Brobdingnag casu-ally refers to the future United States as “our plan-tations in America”.) Suppose Gulliver encounters agiant who is shaped exactly like Gulliver, except heis twelve times as tall. Then the weight of that giantis times that of Gulliver. Thearea of the cross section of one of the giant’s bones is

times that of Gulliver’s corresponding bone.Hence the ratio of the giant’s weight and the area ofhis bone’s cross section is times thatof Gulliver’s corresponding ratio. What do you thinkwould happen to the giant?

39.(10 pts) In a well known story the inventor ofthe game of chess was asked by his well pleasedKing what reward he desired. ”Oh, not much, yourmajesty”, the inventor responded, ”just place a grainof rice on the first square of the board, 2 on the next,

4 on the next, and so on, twice as many on eachsquare as on the preceding one. I will give this riceto the poor.” (For the uninitiated, a chess board has64 squares.) The king thought this a modest requestindeed and ordered the rice to be delivered.Let f � n denote the number of rice grains placed onthe first n squares of the board. So clearly, f � 1 � 1,f � 2 � 1 � 2 � 3, f � 3 � 1 � 2 � 4 � 7, and so on.How does it go on? Compute the next two values off � n : f � 4 � f � 5 �Ponder the structure of this summation and then enteran algebraic expression that definesf � n � as a function of n.Supposing that there are 25,000 grains of rice in apound, 2000 pounds in a ton, and 6 billion peopleon earth, the inventor’s reward would work out to ap-proximately tons of rice for every person onthe planet. Clearly, all the rice in the kingdom wouldnot be enough to begin to fill that request. The storyhas a sad ending: feeling duped, the king caused theinventor of chess to be beheaded.


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Tom Robbins.


due 10/21/02 at 11:59 PM.This homework covers the following topics:� Problems 1-12. Complex Numbers. There

is just one key concept: i acts like any vari-able, except that

i2 �� 1 �� Problem 13-32. Quadratic Equations.This is one of the most central areas of thisclass. I recommend that you solve quadraticequations by completing the square. Thisprocess is based on the binomial formulas.It is superior to using the quadratic formulasince most people are unable to memorize itreliably or apply it accurately.� Problems 33-36 are word problems lead-ing to quadratic equations. Some of theminvolve the Pythagorean Theorem.� Problem 37 is meant to help you prepare forExam 2. It contains a detailed description ofthe problems you should expect on that exam.

1.(10 pts)Let u � 5 � 6i and v � 3 � 2i. Enter the real and imag-inary parts of the following expressions in the appro-priate boxes.u � v � � iu � v � � iu � v � � iu � v � � i

2.(10 pts) This problem is like the preceding one.Let u � 2 � 1i and v � 4 � 3i. Thenu � v � � iu � v � � iu � v � � iu � v � � i

3.(10 pts) This problem is like the preceding one,except that the real or imaginary part may be nega-tive. Let u � 2 � 5i and v �� 4 � 1i. Then

u � v � � iu � v � � iu � v � � iu � v � � i

4.(10 pts) Let u �� 6 � 1i and v � 2 � 3i. Thenu � v � � iu � v � � iu � v � � iu � v � � i

5.(10 pts) Complete the following equations:i3 � � ii4 � � ii5 � � ii6 � � ii2001 � � i

6.(10 pts) Complete the following equations:� 7 � 2i 2 � � i� 7 � 2i 3 � � i7.(10 pts) Let

u � 6 � 8i � v � 6 � 6i � w �� 4 � 2i �Consider the equation

ux � v � w �Solve it for x using exactly the same ideas we used forsolving linear equations with real coefficients. Askwhat bothers you, and get rid of it by doing the samething on both sides of the equation.x � � i

8.(10 pts) This is much like the preceding problem.Let

u �� 4 � 3i � v � 7 � 1i � w � 5 � 8i �Solve the following equation for x:

ux � v � w �x � � i

9.(10 pts) Complete the following equation. Youranswers will be algebraic expressions.� a � bi 2 � � i

10.(10 pts) Complete the following equation. Youranswers will be algebraic expressions.� a � bi 3 � � i

11.(10 pts) Complete the following equations.Your answers will be algebraic expressions.� a � bi �� a � bi � � i� a � bi 2 � � a � bi 2 � � i

1

12.(10 pts) Let u � a � bi and v � c � di. Com-plete the following equations. Your answers will bealgebraic expressions.u � v � � iu � v � � iu � v � � iu � v � � i

13.(10 pts) In the first few problems, fill in theblanks to make a perfect square. For example, in

x2 � 6x � � � x � 2fill in 9 and 3 since

x2 � 6x � 9 � � x � 3 2 �x2 � 14x � � � x � 2.x2 � 8x � � � x � 2.x2 � 20x � � � x � 2.

14.(10 pts) This is like the preceding problem, ex-cept that your answers may be fractions.x2 � 3x � � � x � 2.x2 � 5x � � � x � 2.x2 � 1

3x � � � x � 2.

15.(10 pts) This is like the preceding problem, ex-cept that your answers may be negativex2 � 7

3x � � � x � 2.x2 � 5

7x � � � x � 2.x2 � 2

3x � � � x � 2.

16.(10 pts) This is like the preceding problem, ex-cept that you have to factor out the leading coeffi-cient. For example, in

3x2 � 3x � � 3 � x � 2fill in 3

4 and 12

3x2 � 3x � 34 � 3 � x2 � x � 1

4 � 3 � x � 1

2 2 �

6x2 � 7x � � 6 � x � 2.4x2 � 2x � � 4 � x � 2.

17.(10 pts) This is like the preceding problem.3x2 � 9x � � 3 � x � 2.9x2 � 10x � � 9 � x � 2.

18.(10 pts) This is like the preceding problems ex-cept that your answer is an algebraic expression.x2 � 2sx � � � x � 2.

19.(10 pts) This is like the preceding problems ex-cept that your answer is an algebraic expression.x2 � bx � � � x � 2.

20.(10 pts) In the next few problems of this set youare asked to solve quadratic equations. These areof the form

ax2 � bx � c � 0 �There are usually two solutions that are eitherof the form r 1 s or of the form r 1 si wherei2 �2� 1, and r and s are real numbers. En-ter r and s. Also enter “i” if the solution is aconjugate complex pair of numbers, “1” if bothsolutions are real, or “0” if there is only one real so-lution. In the last case, also enter s � 0.For example, the equation

x2 � x � 1 � 0

has the solution

x �� 1 � 2 1 � 32

i �Enter � 1 � 2, sqrt � 3 � 2, and i here:x � 1 here.The equation

x2 � x � 1 � 0has the solution

x �� 1 � 2 1 � 52�

Enter � 1 � 2, sqrt � 5 � 2, and 1 here:x � 1 here.The equation

x2 � x � 14 � 0

only has the solution

x � � 1 � 2 �Enter � 1 � 2, 0, and 0 here:x � 1 here.

21.(10 pts) The equation

x2 � 2x � 8 � 0

has the solution x � 1 .22.(10 pts) The equation

x2 � 8x � 32 � 0

has the solution x � 1 .2


x2 � 8x � 16 � 0


1 � 3x2 � 3 � 1x � 7 � 7 � 0


1 � 9x2 � 5x � 1 � 2 � 0

has the solution x � 1 .26.(10 pts) Quadratic equations do not always oc-

cur in standard form. Sometimes they have to be con-verted to standard form using our basic principle ofdoing the same thing on both sides to get where wewant to go.

The equation

x � 1x � 2

has only one solution. It isx � .

27.(10 pts) The equation1 � x

x � x

has two real solutions solution. They arex � 1 .

28.(10 pts) The equationx � 1

2x � 1 � x � 1x � 1

has two real solutions. Enter the smaller one hereand the larger one here .


x4 � 10x2 � 9 � 0

has four solutions. Enter them in increasing order:x1 �x2 �x3 �x4 �


x � � x � 2 � 0

has the solutionx � .

31.(10 pts) There are two solutions of the equation

bx2 � cx � a2 � 0

(where a, b, and c are constants, and x is the un-known). They differ by the sign of the square root.Enter the one with the plus sign here

.32.(10 pts) This problem is similar to the preceding

one.There are two solutions of the equation

bx2 � bx � a2 � 0

(where a and b are constants, and x is the unknown).They differ by the sign of the square root. Enter theone with the plus sign here

.33.(10 pts) The height of a triangle is 8 inches less

than its base. The area of the triangle is 192 squareinches. The height of the triangle is inches andthe base of the triangle is inches.

34.(10 pts) The height of a triangle is 25 inchesgreater than its base. The area of the triangle is 625square inches. The height of the triangle isinches and the base of the triangle is inches.

35.(10 pts) You are approaching the island ofHawaii in a small boat. The highest point on Hawaiiis Mauna Loa at 13,677 feet. You see it just barelyabove the horizon. The radius of Earth is 3,963 miles.Ignoring atmospheric effects, you figure that you are

miles in a straight line from the top of MaunaLoa.

36.(10 pts) You are on a pleasure cruise through theuniverse and you crash in the ocean of an unknownplanet. Your spaceship floats on the water and its topis 26 feet above the surface of the water. You swimaway from the spaceship until you see its top on thehorizon. Your laser range meter tells you that youreyes are 2.8 miles away from the top of the spaceship. (You are a capable - if reckless and curious -swimmer.) The radius of the planet is miles.It’s a small world, but it’s all yours. You figure thatthe surface of the earth istimes as large as the surface of your planet, but still,your planet is plenty big enough for you (if only youcan find land somewhere).

3

37.(1 pt) The purpose of this problem is to helpyou prepare for Exam 2. There will be 10 problemson the exam, of the following types:� Solving linear equations , for example

x � 1x � 2 � x � 3

x � 4�

Multiply with both denominators on bothsides, cancel the x2 term, and solve the re-sulting linear equation. The solution of thisparticular equation is .� Solving inequalities. You process inequali-ties like equalities, by doing the same thingon both sides of the inequality. The main dif-ference is that when you multiply with a neg-ative number you reverse the inequality. Forexample, the inequality

3x � 2 � 5x � 1

can be solved by subtracting 5x and 2 on bothsides, followed by a division by � 2, givingx � . Note that the inequality sign havebeen reversed.� Solving Equations involving absolute values.The basic principle is that every absolutevalue may equal the expression of which youtake the absolute value, or its negative. Forexample, � x � 2 � may equal x � 2 or � � x � 2 .So you consider all possible combinations ofthe absolute values, and if there are a lot ofthem you reduce the number of cases judi-ciously. For example, the equation

� x � 1 � � � 2x � 1 �has the obvious solution x � 0 and the lessobvious solution x � .� Drawing the graphs of various forms ofequations defining straight lines. For exam-ple, the graph of the equation

x � y � 1 � 0

is given in the accompanying Figure:

� Given information about a line, like a pointon the line and the slope of the line, draw thegraph, and write down its equation in somespecified form, like the slope intercept form.You can practice this kind of thing by goingto this interactive page .� Computing the intersection of two lines,graphically and algebraically. Algebraicallyyou can do this by writing the point-interceptforms of the two lines, equating the right handsides, solving for x, and evaluating to obtainy. You did this in past WeBWorK problemswhere you computed the distance of a pointfrom a line. Go back to those problems to re-fresh your memory if necessary.� The concept of functions , particularly eval-uating functions at numbers and algebraic ex-pressions. For example if

f � x � 1x � 1

then

f � f � x � � 11 � 1

x � 1

which of course can be simplified.� Evaluating powers like 64 � 56 � .� Combining complex numbers using the

usual complex arithmetic, as practiced onseveral past WeBWorK Problems.� A word problem involving scaling . Re-member, the key idea is that if you multiplyevery linear dimension of an object with acertain factor then you multiply the area ofthe volume with the square of that factor, andthe volume (and weight) with the cube of thatfactor.

4


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Tom Robbins.


due 10/28/02 at 11:59 PM.This set contains a few word problems leading to

quadratic equations now that you know how tosolve them, by applying the quadratic formula , or,more reliably, by completing the square.

All of these problems (1–6) involve objects mov-ing under the influence of gravity, as explained on thepage on throwing rocks and illustrated here .

The focus of the set, however, is on polynomialsand how to work with them. The problems cover thefollowing topics:� Evaluating a polynomial, problem 7–9. A

polynomial can of course be evaluated likeany other function, but there is a particularlyefficient way called nested multiplication.� The language of polynomials, problems10–13, 15, 23, 25-26. As everywhere inmathematics, you need to understand the lan-guage of polynomials to deal with them ef-fectively.� Converting polynomials to standard form ,problems 14, 16–19 This is a task that arisesin many applications of polynomials.� Evaluating a polynomial , problems 20-22at non-numerical expressions. This is likesome similar problems that you did earlier forother functions .� More on the concept of polynomials, prob-lems 24, 27–31. You can answer these ques-tions by understanding and applying the defi-nitions and languyuage of polynomials.� A couple of simple word problems involvingpolynomials, problems 32–33.� A more involved word problem, problem34.

1.(10 pts) Recall that the height h � t at the time tof a rock tossed into the air at time 0 from a height h0

at an initial velocity v0 is given by

h � t � h0� v0t � 16t2 �

Time is measured in seconds, height in feet, and ve-locity in feet per second. The positive direction is up,so if the rock is moving down then its velocity is neg-ative. The magic number 16 in this equation is due tothe mass and radius of earth and would be differentfor example on Mars or on the Moon. The velocity ofthe rock at time t is given by

v � t � v0 � 32t �Suppose you throw a rock upward from a height of

64 feet with an initial velocity of 48 feet per second.The rock will hit the ground after seconds.

2.(10 pts) Suppose that as you throw the rock in thepreceding question, you also give it a forward motionof 20 feet per second. The rock will hit the groundfeet away from you, measured horizontally.

3.(10 pts) You are flying in an open plane at an al-titude of 6400 feet and you drop a Coca Cola bottleout of the window. The bottle will hit the ground after

seconds. (Note: this problem is a bit unrealisticsince it ignores air drag. The scenario described hereprovides the opening scene in the movie ”The Godsmust be crazy.”)

4.(10 pts) You fire a rifle straight up. Your bulletleaves your gun at a velocity of 928 feet per second.Ignore air resistance. Consider the muzzle of yourgun to be at height zero. The bullet will reach itsmaximum altitude of feet after seconds.

5.(10 pts) You fire a rifle at an angle of 45 degrees.Thus the initial horizontal and vertical velocities ofyour bullet are the same. Suppose they each equal320 feet per second. Again ignore air resistance. As-sume you are shooting from ground level (height 0).Your bullet will hit the ground feet from yourcurrent position.

6.(10 pts) You drop a rock into a deep well. Youcan’t see the rock’s impact at the bottom, but you hearit after 7 seconds. The depth of the well is feet.

Ignore air resistance. The time that passes afteryou drop the rock has two components: the time ittakes the rock to reach the bottom of the well, andthe time that it takes the sound of the impact to travel

1

back to you. Assume the speed of sound is 1100 feetper second.

7.(10 pts) Let the polynomial p be defined by

p � x � 2x3 � 3x2 � 4x � 5

Thenp � 1 � ,p � 2 � , andp � � 1 � .

8.(10 pts) Let the polynomial p be defined as in thepreceding problem by

p � x � 2x3 � 3x2 � 4x � 5

Suppose you evaluate the polynomial at x � 2 us-ing synthetic division (also called nested multipli-cation or Horner’s scheme). You obtain an arraythat has three rows, with four entries in the first andthird rows, and three in the second. The entries in thefirst row are (from left to right):

, , , and ,The entries in the second row start in the second col-umn, and are, from left to right:

, , and .The entries in the third row are from left to right:

, , , and .9.(10 pts) This is like the preceding problem. Let

the polynomial p again be defined by

p � x � 3 � 1x4 � 2 � 1x3 � 3 � 2x2 � 1 � 7x � 1 � 4Thenp � � 2 � 3 � .


p � x �� 4x3 � 2x2 � 3x � 5 �The degree of p is ,its leading coefficient is ,and its constant term is ,


p � x �� 10x3 � 2x2 � 3x �The degree of p is ,its leading coefficient is ,and its constant term is ,


p � x � � x � 6 �� x � 9 ��

The degree of p is , its leading coefficient is, and its constant term is ,


p � x � � 5x � 4 �� 8x � 7 3�The degree of p is ,its leading coefficient is ,and its constant term is ,


p � x � � x � 10 � x � 9 3�Then p � x � x2 � x + ,


p � x � � x � 1 � x � 2 � x � 3 � x � 4 The degree of p is ,its leading coefficient is ,and its constant term is ,


p � x � � x � 2 � x � 5 � x � 9 ��Then p � x � x3 � x2 � x � ,Note: some of the coefficients may be negative.

17.(10 pts) This problem is like the preceding one,except you have to get all answers right before receiv-ing credit. Let the polynomial p be defined by

p � x � � x � 3 � x � 5 � x � 9 ��Then p � x � x3 � x2 � x � ,Note: some of the coefficients may be negative.


p � x � � x � 1 � x � 1 3�Thenp � x � x2 � x �Note: some of the coefficients may be negative.


p � x � � x � 1 � x2 � x � 1 ��Thenp � x � x3 � x2 � x �Note: some of the coefficients may be negative.


p � x � x3 � 2x2 � 3x � 4 �Then p � x � 1 � x3 � x2 � x � ,

2

21.(10 pts) This is like the preceding problem ex-cept that you must get all answers right before receiv-ing credit. Let the polynomial p be defined by

p � x � x3 � 2x2 � 3x � 4

Then p � x � 1 � x3 � x2 � x � ,22.(10 pts) This is like the preceding problem. Let

the polynomial p be defined by

p � x � x3 � 2x2 � 3x � 4 �Then p � 2x � 1 � x3 � x2 � x � ,

23.(10 pts) This may be a little more challenging,but you do not need to expand this polynomial tostandard form! Let the polynomial p be defined by

p � x � � x � 2 17 �The degree of p is ,its leading coefficient is ,and its constant term is ,

24.(10 pts) Indicate with true (T) or false (F)whether the following functions are polynomials.

f � x � x � 2 �f � x � � 2 �f � x � � x �f � x � � x ��f � x � 1

x �f � x � � x � 1 200000 �

f � x � � 1 � x2 �25.(10 pts) Match the verbal descriptions with the

given polynomials. You need to use all polynomi-als and all descriptions. Recall that polynomials ofdegrees 0, 1, 2, 3, 4, 5, are called constant, linear,quadratic, cubic, quartic, and quintic, respectively.Also recall the definitions of the terms monomial, bi-nomial, trinomial, given here .You must get all of the answers correct to receivecredit.

1. The square of a cubic polynomial2. A cubic polynomial3. A trinomial4. A quartic binomial5. A quintic monomial.A. πx5

B. x4 � 2x3

C. x3 � 3x2 � 3x � 1D. x2 � 2x � 1

E. � x3 � 1 226.(10 pts) This is much like the preceding prob-

lem. You may have to manipulate the algebraic ex-pressions defining the polynomials to recognize thecorrect match.You must get all of the answers correct to receivecredit.

1. A trinomial2. A quintic monomial.3. A cubic polynomial4. The square of a cubic polynomial5. A quartic binomial

A. πx5

B. x3 � x � 2 C. � x � 1 3D. � x � 1 2E. x6 � 2x3 � 1

27.(10 pts) Think about the following statementsand indicate whether they are true (T) of false (F).You need to get all answers correct before obtainingcredit.

The product of 2 linear polynomials isquadratic.

The sum of two cubic polynomials cannot havea degree greater than 3.

The sum of two cubic polynomials may have adegree less than 3.

The sum of a cubic and a quartic polynomialmay have a degree different from 4.

The product of two monomials is a monomial.The product of two binomials is a binomial.

28.(10 pts) Think about the following statementsand indicate whether they are true (T) or false (F).You need to get all answers correct before obtainingcredit.

The graph of a linear polynomial is a straightline.

The degree of a trinomial is at least 2.The product of two polynomials is always a

polynomial.The quotient of two polynomials is always a

polynomial.The sum of two polynomials is always a

polynomial.3

The difference of two polynomials is always apolynomial.

29.(10 pts) Suppose you multiply a polynomial ofdegree m with a polynomial of degree n. The result isa polynomial of degree

30.(10 pts) Suppose you add a polynomial of de-gree m to a polynomial of degree n where m � n.Then the result is a polynomial of degree

31.(10 pts) Suppose you subtract a polynomial ofdegree m from a polynomial of degree n where m � n.Then the result is a polynomial of degree

32.(10 pts) The obelisk in the movie 2001 has theshape of a rectangular box with lengths x, 4x, and 9x,where x is a parameter.The volume V of the obelisk is a polynomial expres-sion in x of degree and leading coefficient.In fact, V � (enter an expression in x).

33.(10 pts) The area A of that same obelisk is apolynomial expression in x of degree andleading coefficient .

In fact, A � (enter an expression in x).

34.(10 pts) Here is a small challenge for your en-tertainment and gratification. For any natural numbern, let f � n denote the sum of the numbers from 1 to n.Thus f � 1 � 1, f � 2 � 1 � 2 � 3, f � 3 � 1 � 2 � 3 �6, f � 100 � 1 � 2 � 3 � �� 100 � 5050, etc.It turns out that f is a polynomial of degree 2 in n.Figure out the coefficients of f :f � n � n2 � n + ,There is a story about Carl Friedrich Gauss (1777-1855) who may have been the most outstandingmathematician in human history. According to thestory, when Gauss was seven years old, his teacher atone stage was unhappy with the class and as a punish-ment he asked them to compute f � 100 . Gauss’ classmates started writing the numbers from 1 to 100 ontheir paper, and adding those numbers. Gauss staredat the ceiling and then wrote the single number 5050on the sheet and handed it in. You aren’t Gauss, butyou also aren’t seven years old, so maybe you canfigure out what he was thinking!


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Tom Robbins.


due 11/4/02 at 11:59 PM.This set has fewer problems than usual, but there

are a couple that may be a little more time consum-ing than usual.

The focus on the set is on factoring polynomialsusing synthetic and long division, and utilizingknowledge about the roots of a polynomial to iden-tify its factors. An application of these techniques isthe Euclidean Algorithm .

1.(10 pts) Use synthetic division to divide a poly-nomial with remainder.

x2 � 2x � 4 � � x � 1 � � � � 2.(10 pts) Use synthetic division to divide these

two polynomials with remainder:x2 � 8x � 15 � � x � 1 � � � �

x2 � 14x � 49 � � x � 4 � � � � 3.(10 pts) Use synthetic division to divide these

two polynomials with remainder:x2 � 8x � 1 � � x � 1 � � � � x2 � 1x � 35 � � x � 6 � � � �

4.(10 pts) Use synthetic division to divide thesetwo polynomials with remainder:x2 � 1x � 12 � � x � 1 � � � � x2 � 8x � 17 � � x � 2 � � � �

5.(10 pts) This is like the preceding problems ex-cept that you divide by a quadratic term and obtain alinear remainder. Use long division to divide thesetwo polynomials with remainder:

2x4 � 11x3 � 19x2 � x � 29 � � x2 � 4x � 6 � � � � 6.(10 pts) Use long division to divide these two

polynomials with remainder:6x4 � 2x3 � 3x2 � 20x � 22 � � 3x2 � 2x � 5 � � � �

7.(10 pts) Find the greatest common factors of thepairs of numbers below. I recommend you use theEuclidean Algorithm .gcf � 301 � 217 �gcf � 1829 � 1711 �gcf � 493 � 203 �

8.(10 pts) Find the greatest common factors of thepairs of numbers below. I recommend you use theEuclidean Algorithm .gcf � 82519 � 90817 �gcf � 94417 � 131753 �gcf � 64507 � 49601 �

9.(10 pts) In this problem you are given two poly-nomials p and q and you are asked to find their great-est common factor. I recommend that you use theEuclidean Algorithm. (It’s really simple in this case.)After you compute the common factor divide it by asuitable integer so that the leading coefficient is 1.

p � x � x4 � 2x3 � 6x2 � 8x � 8 �q � x � x4 � 2x3 � 8x2 � 8x � 16 �

The greatest common factor (i.e., the one of thehighest degree) of p and q is .Using your new found knowledge simplify the ratiop 4 x 5q 4 x 5 � � �� .

10.(10 pts) The polynomial

p � x � x3 � 6x2 � 11x � 6

has three real roots. List them in increasing sequence:, , .


p � x � x3 � 3x2 � x � 1

has three real roots. List them in increasing sequence:, , .


p � x � 24x3 � 46x2 � 29x � 6

has three real roots. One of them is x � 23 Find the

others and list them in increasing sequence:and .


p � x � 3x3 � 10x2 � 5x � 21

has three real roots. One of them is x �6� 23 Find the

others and list them in increasing sequence:and .


p � x � 4x3 � 16x2 � 21x � 27

has the real root x � 3 and a conjugate complex pairof roots which is 1 i.


p � x � x4 � 6x3 � 15x2 � 18x � 10

four complex roots. One of them is x � 1 � i. Thisimmediately tells you another root, namely� i. The other conjugate complex pair ofroots is 1 i.


p � x � 36x4 � 108x3 � 245x2 � 432x � 404

has four complex roots. One of them is x � 2i. Thisimmediately tells you another root, namely� i. The other conjugate complex pair ofroots is 1 i.


p � x � x5 � x4 � 13x3 � 13x2 � 36x � 36

has five real roots. One of them is x � 1. List theother four in increasing sequence:

, , , and .18.(10 pts) Consider the the polynomial

p � x � x3 � � a � 1 x2 � � a � 6 x � 6a

where x is the variable and a is a parameter. (Thatmeans we think of it as a particular real number, butwe don’t specify which.) It is easy to check thatp � a � 0 �p has two more real roots. They are

and (in increasing sequence).19.(10 pts) A boat travels at a speed of 20 miles

per hour in still water. It travels 48 miles upstream,and then returns to the starting point in a total of fivehours. The speed of the current is miles perhour.

20.(10 pts) The speed of a commuter plane is 150miles per hour slower than that of a passenger jet.

The commuter plane travels 450 miles in the sametime the jet travels 1150 miles. The speed of the com-muter plane is miles per hour and that of the jetis miles per hour.

21.(10 pts) You have an aquarium that is one foothigher than it is wide, and one foot longer than it ishigh. It holds 6 cubic feet of water. The width of theaquarium is feet, its height is feet, and itslength feet.

22.(10 pts) This problem is a little more challeng-ing. Most exercises in a class like this one can bedone very quickly. However, the values of the tech-niques we learn in this and other basic classes lies intheir being building blocks for the solutions of com-plicated problems. Here is a problem that requiresquite a few steps, each of which is one that we havepracticed in this class. But we have to put them alltogether to reach the final answer.Let

f � n � 12 � 22 � 32 � � �� n2 �Thus

f � 1 � 12 � 1 �f � 2 � 12 � 22 � 5 �f � 3 � 12 � 22 � 32 � 14 �f � 4 � 12 � 22 � 32 � 42 � 30 �

etc.It turns out that f is a polynomial of degree 3 in n.Figure out the coefficients of f :

f � n � n3 � n2 � n � ,

One way to solve the problem is described in thisHint: Write

f � n � a� b � n � 1 � c � n � 1 � n � 2 � d � n � 1 � n � 2 � n � 3 where a, b, c, d are constants that we still have to de-termine. The significance of writing f in this form isthat when n � 1 only the first term is non-zero, whenn � 2 only the first two terms are non-zero, and soon. Note that f � 1 � a, and so a must be 1 (since theterms involving b, c, and d are zero). Knowing a, askwhat f � 2 is, use that value to figure out b, and goon from there. Once we know a, b, c and d, we canconvert f to standard form.

2


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Tom Robbins.


due 11/11/02 at 11:59 PM.This set deals with rational expressions and functions.

You can think of a rational expression as the ratio oftwo polynomials or as an expression that can beevaluated using only the four basic arithmetic op-erations. The one basic principle in manipulatingrational expressions is that they work exactly likefractions except. (Now you know why this classstarted out with fractions; they are not obsolete de-spite the modern ubiquity of calculators.)

Many of the questions ask you to enter the numer-ator and denominator after canceling any commonpolynomial or integer factors. Note that you can en-ter polynomial expression in non-standard, e.g., fac-tored, form. In some cases the cancellations of cer-tain factors in numerator and denominator impliesthat some values of x are inadmissible (since they ren-der the factor zero). If you are working on paper youwould take a note. As far as this assignment is con-cerned ignore the issue in most problems. (A coupleask when the denominator is zero.) Enter a denomi-nator 1 if your answer is a polynomial, and multiplynumerator and denominator with � 1 if the leadingcoefficient of your denominator is negative.

A few examples show some of the simplifications:Example 1.

2x � 44x � 6 � 2 � x � 2

2 � x � 3 � x � 2x � 3

�Thus you would enter the polynomial expressionsx � 2 and x � 3 as your solution. In this case youcanceled a common integer factor 2 in numerator anddenominator.

Example 2.

x � 2� 3x � 2 � � � 1 � � x � 2 � � 1 �� 3x � 2 � � x � 23x � 2

�

In this case we are not really simplifying the ex-pression, but we are making the leading coefficientof the denominator positive (by multiplying with � 1in numerator and denominator), to accommodate theidiosyncrasies of WeBWorK. You would enter � x � 2and 3x � 2 as your answers.

Example 3.

x2 � 5x � 6x2 � 6x � 8 � � x � 2 �� x � 3 � x � 2 �� x � 4 � x � 3

x � 4�

You would enter x � 3 and x � 4 as you answers, andif the problems asked for what value of x the iden-tity does not hold you’d say x �6� 2 because for thatvalue the original expression is not defined, but thisis not apparent in the final expression. (None of theexpressions is defined when x � � 4 but that does notneed to be noted since in that case the denominator isobviously zero.)

1.(10 pts) Simplify the expression 15x � 219x � 15 � � �� .

2.(10 pts) Simplify the expression x � 1� x � 1 � � �� .3.(10 pts) Simplify the expression x2 � 2x

x2 � x � � �� .4.(10 pts) Cancel common polynomial and integer

factors. and fill in the blanks.x2 � 10x � 24x2 � 12x � 32 � � x � 7�� x � For this identity to hold, x must not equal .

5.(10 pts) Cancel common polynomial and integerfactors. and fill in the blanks.x2 � 11x � 30x2 � 13x � 42 � � x � 8�� x � /For this identity to hold, x must not equal .

6.(10 pts) Simplify the expression x2 � 1x � 1 � � �� .

7.(10 pts) Simplify the expression x3 � 2x2 � x � 2x2 � 5x � 6 � � �� .

8.(10 pts) Simplify the expression x4 � 1x4 � 2x2 � 1 � � �� .

9.(10 pts)1

x � 1� 1

x � 6 � � �� .1

x � 1 � 1x � 6 � � �� .

1

1x � 1� 1

x � 6 � � �� .1

x � 1� 1

x � 6 � � �� .10.(10 pts)

2x � 1� 1

x � 8 � � �� .2

x � 1 � 1x � 8 � � �� .

2x � 1� 1

x � 8 � � �� .2

x � 1� 1

x � 8 � � �� .11.(10 pts)

15x � 11� 11

x � 12 � � �� .15

x � 11 � 11x � 12 � � �� .

15x � 11

� 11x � 12 � � �� .

15x � 11� 11

x � 12 � � �� .12.(10 pts)1

x2 � 36� 1

x � 6 � � �� .1

x2 � 36 � 1x � 6 � � �� .

1x2 � 36

� 1x � 6 � � �� .

1x2 � 36

� 1x � 6 � � �� .

13.(10 pts)1

x2 � 4x � 4� 1

x � 2 � � �� .1

x2 � 4x � 4 � 1x � 2 � � �� .

1x2 � 4x � 4

� 1x � 2 � � �� .

1x2 � 4x � 4

� 1x � 2 � � �� .

14.(10 pts)1

x2 � 10x � 25� 1

x � 5 � � �� .1

x2 � 10x � 25 � 1x � 5 � � �� .

1x2 � 10x � 25

� 1x � 5 � � �� .

1x2 � 10x � 25

� 1x � 5 � � �� .

15.(10 pts)1

x2 � 13x � 36� 1

x � 4 � � �� .1

x2 � 13x � 36 � 1x � 4 � � �� .

1x2 � 13x � 36

� 1x � 4 � � �� .

1x2 � 13x � 36

� 1x � 4 � � �� .

16.(10 pts)1

x � 1� 1

x � 21

x � 2� 1

x � 3� � �� .

17.(10 pts) Let

f � x � 1x�

Thenf � x � 1 � � �� . andf � f � x � � � �� .

18.(10 pts) Let

f � x � 1x � 1

�Thenf � x � 1 � � �� .and f � f � x � � ��

19.(10 pts) Let

f � x � xx � 1

�Thenf � x � 1 � � �� , andf � f � x � � � ��

20.(10 pts) Let

f � x � x � 1x � 1

�Thenf � f � x � � � ��

21.(10 pts) In order for the identity1

x � 1� a

x � 1 � � 2x2 � 1

to hold for all x, a must equal .22.(10 pts) In order for the identity

1x � 1

� ax2 � 1 � x

x2 � 1to hold for all x, a must equal .

23.(10 pts) In order for the identitya

x � 1� b

x � 1 � 1x2 � 1

to hold for all x,a must equal andb must equal .

24.(10 pts) (This is essentially problem 92 on page271 of the textbook. Your answer is an algebraic ex-pression involving d).A circular swimming pool has a depth d and a radiusthat equals five times its depth. A rectangular pool

2

has the same depth as the circular pool. Its width isfour feet more than three times its depth and its lengthis two feet less than six times its depth. The ratio ofthe rectangular pool’s volume to the circular pool’svolume is .

25.(10 pts) (This is essentially problem 90 on page271 of the textbook. Most of your answers are alge-braic expressions involving t.)A car starts on a trip and travels at a speed of 55 mph.Two hours later, a second car starts on the same tripand travels at a speed of 65 mph.When the second car has been on the road for t hours,the first car has traveled miles and the secondcar has traveled miles.At time t the distance between the first car and thesecond car is miles.The ratio of the distance the second car has traveledand the distance the first car has traveled is .The second car catches up with the first carhours after the departure of the first car. (Those aresome determined drivers!)

26.(10 pts) When two resistors S and T are con-nected in parallel their combined resistance R is givenby the expression

R � 11S� 1

T

�

This expression can be rewritten as a rational expres-sion in S and T . The numerator of that expression is

and its denominator is

27.(10 pts) Suppose you connect two resistors ofthe same resistance in parallel. Then the resistance ofthe two connected resistors equals times that ofeach resistor alone.

28.(10 pts) I found the following definition in TheHarper Collins Dictionary of Mathematics:“golden mean, golden section, or extreme andmean ratio, n. the proportion of the division of a lineso that the smaller is to the larger as the larger is tothe whole, or of the sides of a rectangle so that the ra-tio of their difference to the smaller equals that of thesmaller to the larger, supposed in classical aesthetictheory to be uniquely pleasing to the eye.”Those Greek were onto something! Apparently that“extreme and mean ratio” is a specific number. In factit is . By the way, don’t get putoff by the above piece of language which is opaqueonly because of its brevity. The Harper Dictionary ofMathematics is excellent, despite occasional idiosyn-crasies. It’s also an inexpensive paperback that I usealmost every day, and that I recommend highly as anaddition to your library.


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due 11/18/02 at 11:59 PM.One of the major themes in Math 1010 is the use ofCartesian Coordinates to provide a link betweengeometry and algebra, thereby greatly increasing ourproblem solving ability. The graph of an equation in-volving the variables x and y is the set of all points� x � y whose coordinates satisfy the equation. Thegraph of a function f is the graph of the equationy � f � x . Simple algebraic modifications of the equa-tion have simple effects on the graph of the equation.In this class we illustrate these concepts in the con-text of parabolas and other Conic Sections, but theprinciples apply much more generally.

The web pages for this course contain aninteractive page that lets you explore the interplaybetween geometry and algebra.

A major tool in recognizing the graph ofsome of the functions involved is the process ofcompleting the square.

Let’s see how this works in the case of a parabola.Suppose we are given the quadratic function

f � x � 2x2 � 4x � 5 �We want to write it in the form

f � x � a � x � h 2 � k �Thus we need to figure out a, h, and k, as follows:

f � x � 2x2 � 4x � 5 � factor out 2� 2 � x2 � 2x � 5 � add and subtract 2� 2 � x2 � 2x � 1 � 2 � 5 � complete the square� 2 � x � 1 2 � 3 � a � 2 � h � 1 � k � 3 �This pattern is repeated in many of the problems.

A couple deal with the general case where the coef-ficients involved are variables themselves rather thanspecific numbers.

The last problem on this set is a survey of Exam3. As usual, work through it, make sure you under-stand every part of it, and let me know if you haveany questions.

There are a couple of problems on the exam thatdeal with very simple linear systems which will bethe main subject of home work 13.

1.(10 pts)Indicate whether the following equations define a

straight line (L), a circle (C), an ellipse (but not a cir-cle) (E), a parabola (P), or a hyperbola (H). Remem-ber that WeBWorK is case sensitive, so you must en-ter capital letters L, C, E, P, H for your answer to berecognized. To get you going, on this and the nextproblem WeBWorK will tell you about each individ-ual answer whether it is correct, but make sure youunderstand what you are doing so that you are pre-pared for the next problems.

1. x2 � y2

2 � 12. � x � 1 2 � � y � 3 2 � 43. 3x � 4y � 5 � 04. y �� 3 � x � 1 2 � 45. x2 � y2 � 1

2.(10 pts) In the preceding problem the variousequations were in some sort of standard form. Inthis problem you may have to convert the equationsto standard form first before recognizing what kindof graph they have. Indicate whether the followingequations define a straight line (L), a circle (C), anellipse (but not a circle) (E), a parabola (P), or a hy-perbola (H).

1. 2x2 � y2 � 1 � 02. 3x � 4y � 53. x2 � 4x � y2 � 6y � 9 � 04. y � 2x2 � 3x � 45. x2 � 4x � y2 � 6y � 9 � 0

3.(10 pts) This is like problem 1 except that youmust get all answers correct before getting credit.Indicate whether the following equations define astraight line (L), a circle (C), an ellipse (but not a cir-cle) (E), a parabola (P), or a hyperbola (H).

1. x2

9� y2

2 � 12. 2x � y � 13 � 03. � x � 3 2 � � y � 4 2 � 9

1

4. � x � 1 2 � � y � 1 2 � 15. y � 5 � x � 11 2 � 4

4.(10 pts) This is like problem 2 except you needto get all answers correct before getting credit. Indi-cate whether the following equations define a straightline (L), a circle (C), an ellipse (but not a circle) (E),a parabola (P), or a hyperbola (H).

1. � 2x � y � 172. y2 � 6y � x2 � 6x � 9 � 03. 15x2 � 5y2 � 49 � 04. x2 � 6x � y2 � 8y � 05. y � 2x2 � 3x � 4

5.(10 pts) This is like the preceding problem ex-cept it may be not quite as straightforward to recog-nize that nature of the graphs. Indicate whether thefollowing equations define a straight line (L), a circle(C), an ellipse (but not a circle) (E), a parabola (P), ora hyperbola (H).

1. y2 � � y � 1 2 � x2 � 02. x2 � y2 � 4x � 6y � 93. � x � 1 2 � x2 � y � 04. y2 � � x � 1 2 � 2x2 � 1005. x2 � � 2y � 1 2 � 20

6.(10 pts) Indicate whether the equationyx � 1

defines a straight line (L), a circle (C), an ellipse (butnot a circle) (E), a parabola (P), or a hyperbola (H).Ignore the fact that the ratio on the left is undefinedwhen x � 0.

7.(10 pts) The equation� x � 2 2 � � y � 5 2 � 9

defines a circle with center � � and radius.


x2 � 8x � y2 � 6y � 50 � 0

defines a circle with center � � and radius.

9.(10 pts) The equation� x � 2 29

� � y � 4 225 � 1

defines an ellipse with center � � . Themajor axis has length and the minor axis haslength .


9x2 � 18x � 4y2 � 8y � 23

defines an ellipse with center � � . Themajor axis has length and the minor axis haslength .

11.(10 pts)Consider the circle in the above Figure. The circle

has center � � and radius . Theequation of the circle can be written in standard formas � x � 2 � � y � 2 �

. (All the required answers are integers. Theymay be negative, however.)

12.(10 pts)2

Consider the circle in the above Figure. The equa-tion of the circle can be written in standard form as� x � 2 � � y � 2 �

. (All the required answers are integers. Theymay be negative, however.)

13.(10 pts)Consider the ellipse in the above Figure. The el-

lipse has center � � , its major axis haslength , and its minor axis has length .The equation of the ellipse can be written in standardform as � x � 2 � � � y � 2 � � 1.

14.(10 pts)The equation of the ellipse in this Figure can be

written in standard form as � x � 2 � � � y � 2 � � 1.

15.(10 pts) A major skill in mathematics is the abil-ity to apply what you learned to different contexts.So remember what you learned about translations of

circles and parabolas and consider the absolute valuefunction f � x � � x � whose graph is given in the firstFigure in this problem.

You can describe this function to WeBWorK by en-tering abs(x) here: . (Go ahead, try it rightnow.)Consider the function whose graph is given in thisFigure:

Let’s call that function g � x � .(Enter an algebraic expression involving the absolutevalue function abs.)

16.(10 pts) This is similar to the preceding prob-lem. Consider the function whose graph is given inthis Figure:

3

Let’s call that function g � x � .(Enter an algebraic expression involving the absolutevalue function abs.)

17.(10 pts) The next two problems are similar tothe preceding two, except that the basic function isf � x � x3 instead of � x � . Its graph is given in the firstFigure in this problem.

You can describe this function to WeBWorK by en-tering x**3 here: . (Go ahead, try it rightnow.)Consider the function whose graph is given in thisFigure:

Let’s call that function g � x � . (Enter analgebraic expression.)

18.(10 pts) This is similar to the preceding prob-lem. Consider the function whose graph is given inthis Figure:

Let’s call that function g � x � . (Enter analgebraic expression.)

19.(10 pts) In this problem you are asked to convertthe general standard form of a quadratic polynomialinto the completed square form. Suppose

f � x � Ax2 � Bx � C

where A, B, and C are the coefficients of the quadraticpolynomial, and A /� 0. Then f can be written as

f � x � a � x � h 2 � k

where4

a � ,h � , andk � .Enter your answers as algebraic expressions in A, B,and C. Remember that WeBWorK (and mathemat-ics) is case sensitive so in this case your variables arecapital letters, and you must enter them as such.

20.(10 pts) This problem is similar to the preced-ing one, except it deals with a circle. Consider theequation

x2 � y2 � Ax � By � C � 0 �It describes a circle with center � h � k and radius rwhereh � ,k � , andr � .Enter your answers as algebraic expressions in A, B,and C. Remember that WeBWorK (and mathemat-ics) is case sensitive so in this case your variables arecapital letters, and you must enter them as such.

21.(10 pts) The next few problems serve as a warmup for the next homework set which will deal withlinear systems of equations. Consider the system ofequations

x � y � 11x � y � 10

(Do you recognize this system?) Its solution isx � and y � .

22.(10 pts) Consider the system of equations

3x � 2y � 17x � y � � 1

Its solution isx � and y � .

23.(10 pts) Consider the system of equations

3x � 2y � 12x � 3y � 2

Its solution isx � and y � . (I would enter my answers asfractions.)

24.(10 pts) Find two positive integers whose sumis 48, and difference is 12.The smaller is and the larger is .

25.(10 pts) Find two positive integers whose sum is48, and whose sum also equals twice their difference.The smaller integer is

and the larger is .

26.(10 pts) You take a group of 20 people going tothe movies. Kids’ tickets cost $4.- each, and adultstickets $6.- each. Your total cost is $96.- . The num-ber of kids in your group is

27.(1 pt) This problem serves as a preparation forExam 3. Each of the bulleted items in the followinglist corresponds to one of the exam problems. Makesure you thoroughly understand each problem, andthe issues surrounding it, and send me a message ifyou have any questions!� Solving quadratic equations of the form

x2 � bx � c � 0 �You can do this by completing the square, ap-plying the quadratic formula, or in some(rare) cases by factoring. For example, theequation

x2 � 2x � 15 � 0

has two real solutions, the larger one of whichis and the smaller of which is .� Quadratic equations do not always occur inthe form listed above. If x2 is multiplied by afactor as in

4x2 � 12x � 6 � 0

you can still apply the quadratic formula, butif you want to complete the square you firstneed to divide by the leading coefficient firston both sides of the equation. Be careful ifyou decide to rely on the quadratic formula,in my experience few people are able to re-member it dependably and to apply it cor-rectly. The solution of the above equation isx � 1� Sometimes equations do not look like qua-dratic equations but can be converted to such.For example, in the equation

x4 � 13x2 � 36 � 0 ��5

you would think of z � x2 as the variable andobtain a quadratic equation in z:

z2 � 13z � 36 � 0 �Once you know z you can find x by comput-ing the positive and negative square roots ofz. For example, the largest solution of �� is

.In the equation

x � 13�

x � 36 � 0

you think of z � � x as the variable, solve forz, square z and get the largest solution .

Sometimes you have rational expressionsand you obtain a quadratic equation aftermultiplying with the appropriate denomina-tors, as in the equation

x � 1x � x

which has a pair of conjugate complex solu-tions: x � 1i.� In radical equations you get rid of the radicalsby isolating them and taking them to the ap-propriate power. For example, the solution ofthe equation�

x � � x � 1 � 3

is x �

� Polynomials, like any functions, can be eval-uated, at numbers and also at algebraic ex-pressions. For example if

p � x � 3x2 � 4x � 5

then p � x � 3 � x2 - x � .� Understand how to use synthetic division todivide a polynomial by a linear factor.� Understand how to draw the graph of a qua-dratic function. It is always a parabola.� Understand how to obtain the

standard form of a rational function(as a ratio of two polynomials). Consider

for example the function

r � x � 1x � 1 � 1

1 � 1x

�It can be simplified to

r � x � 1 � x1 � x

�the ratio of two polynomials. To get rid of theratio in the denominator of the second termmultiply the numerator and denominator inthe second term with x.

As an exercise simplify the following ex-pression

1x � 1� 1

1 � 1x� .� Understand how to solve a

system of two linear equations in two unknowns.� The last problem is a word problem. It leadsto two linear equation in two unknowns.


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Tom Robbins.


due 11/25/02 at 11:59 PM.The focus of this home work set is onthe solution of linear systems . You can seea large worked example .

There are only 10 problems but the larger of thelinear systems in this set may take you a little longerthan usual.

You want to guard against making mistakes, par-ticularly in the solution of linear systems. I recom-mend that you go about the solution of each systemcarefully and deliberately. Any arithmetic error willnot be immediately apparent and yet it will invalidateall your subsequent work. I recommend you use rowsums to detect errors right when they occur and youorganize your work in such a way that you can easilygo back and see and check what you have done so far.

Note that some of the problems differ only in theright hand side. In those cases you want to processthe left hand side only once, so again take careful noteof what you do.

Here is a hint on elimination. Suppose you havethe two equations

5x � 3y � 27x � 11y � 3

and you want to eliminate x. A brute force way ofdoing this is to divide the first equation by 5 (to makethe coefficient of x equal to 1) and then subtract 7times the new first equation from the second equa-tion. The drawback of this idea is that it requiresfractional arithmetic, which is awkward for hand cal-culation. You can avoid that problem by forming anew equation that is 7 times the first equation minus5 times the second equation, which gives 76y �9� 1 �The solutions that you can see after this set closes usethis trick in a couple of places.

1.(10 pts) As a warm-up solve the following linearsystem:

2x � 3y � 1x � y � 3

The solution of this system isx � and y � .

2.(10 pts) The solution of the linear system

2x � 3y � 3x � y � 1

x � and y � .3.(10 pts) A linear system may have a unique so-

lution, no solution, or infinitely many solutions. Indi-cate the type of the system for th following examplesby U, N, or I, respectively.

1. 2x � 3y � 52x � 3y � 6

2. 2x � 3y � 52x � 4y � 6

3. 2x � 3y � 54x � 6y � 10

4.(10 pts) A linear system may have a unique so-lution, no solution, or infinitely many solutions. Indi-cate the type of the system for th following examplesby U, N, or I, respectively.

1. 2x � 3y � 04x � 6y � 0

2. 2x � 3y � 5x � 6y � 7

3. 2x � 3y � 02x � 4y � 0

4. x � y � 5x � 2y � 10

5. x � 3y � 5x � 3y � 5


2x � 3y � z � 14x � y � z � 0

3x � 2y � 3z � 15

The solution of this system isx � , y � , and z � .Note that the next problem differs from this one onlyin the right hand side. You want to keep careful trackof your calculations in this problem so that you won’t

1

have to redo your processing of the left hand side inthe next problem.


2x � 3y � z � 1x � y � z � 2

3x � 2y � 3z � 3

The solution of this system isx � , y � , and z � .

7.(10 pts) The next two problems are like the pre-ceding two. So keep track of your calculation in thisproblem so that you don’t need to redo them in thenext problem.

The solution of the linear system

x � 2y � 5z � � 322x � 2y � z � 154x � 2y � 3z � 27



x � 2y � 5z � 12x � 2y � z � 24x � 2y � 3z � 3



w � 2x � 3y � 4z � � 52w � x � y � 2z � � 73w � x � 2y � z � 74w � 4x � 2y � 3z � 3

The solution of this system isw � , x � , y � , and z � .

10.(10 pts) The cubic polynomial p satisfying

p � x � 2x for x � 0 � 1 � 2 � 3isp � x � .Enter your solution as an algebraic expression in x.


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Tom Robbins.


due 12/2/02 at 11:59 PM.This is the final home work set for this semester. Its main purpose is to review the entire semester and help

you prepare for the final exam.The last “problem” on this set is a questionnaire asking your opinion about WeBWorK. In our department

we are constantly striving to increase the benefit ow WeBWorK to our students, and we would very muchappreciate if you could complete that questionnaire. The information you provide will be mailed to a centraladdress, tabulated, and distributed to interested parties. It will not be traced back to you personally. (Of course,there will be no questionnaire on the final.)

The final exam in this class will be similar to this home work. However, it will not simply be the same setof problems with different numbers!

You should be able to answer questions like these (but not just these particular questions) on the final exam.To make the best use of this home work I suggest you do the following:� Use the Principles Page and the on-line glossary . Mathematics is partly a language and you are

unnecessarily hampering yourself and wasting time and effort if you don’t make a deliberate effort tounderstand that language. Make sure you have a standard dictionary handy, and if you encounter aword that you don’t fully understand, don’t brush over it but look it up. It only takes a moment, it willsave you time later, and it will increase your understanding and appreciation of what you are reading.� Read through this assignment and decide where you need to review material. For your convenience theproblems contain links to the relevant web pages that you can follow if your bring up the problem online. The questions contain no hints because there won’t be hints in the final either.� When you are ready, but only then, set aside a time where you can answer the questions on this setlike you would if you were taking an exam (lasting 2 full hours). Work the problems on paper, withoutthe aid of notes, books, or calculators, and without consulting people. Do all the things you will doduring the final exam itself. For example, if you get stuck somewhere, set aside the problem and returnto it later after answering the easier questions. When you are through, check and double check youranswers!� For each problem where you did not easily and correctly find the answer, go over your notes, readthe textbook, look in previous exams, home works, and answer sets, and figure out the mathematicsthat surround the problem. Do several problems like the one you got stuck on. There are very manyproblems in the textbook you can use, or you can make up your own.� In addition to the problems on this set, between now and the final exam, do as many word problems asyou can. You can take them from the book, or make them up yourself. Doing so will help you reviewthe material, prepare for the final, and, most importantly, prepare you for future work and studies.� A couple of days prior to the actual final exam, do this problem set again on paper. This will be pastthe deadline for this set, so you will be able to see the correct answers even if you have not alreadyentered them. You will also be able to see detailed solutions after the due date by clicking on ”showsolutions” and then resubmitting your answer. When you are done with your paper work compare your

1

answers with the correct ones. If you still have difficulties answering some of the questions study thesolutions, and go back over your notes. Ask me, Kazuma, or a tutor if you get stuck somewhere.� On the day before the exam, relax, and do something fun and enjoyable. Then approach the actualexam relaxed and confident, knowing that you are ready, and that you can handle this!

Peter Alfeld, JWB 127 or 236, 581-6842.

1.(10 pts) The key to algebra isthat algebraic expressions work just likearithmetic expressions . In particular,rational expressions work just like fractions .That is why it is so important to understand fractions.The problem on this page summarizes the four ba-sic arithmetic operations with fractions, i.e., addition,subtraction, multiplication, and division. Your an-swer should be a simple fraction, with no commonfactors in numerator and denominator. Don’t botherwith mixed numbers .Simplify35� 1

334 � 2

5� /

2.(10 pts)An important task in algebra is to solve an

equation . This means we find a value, or valuesof the variable, or variables that make the equationtrue.

A type of equation that occurs particularly fre-quently are linear equations . The equation in thisproblem is of that type.

The equation

15x � 2 � 85 � 4 � x � 2 has the solution x �

3.(10 pts) Another major type of equation arequadratic equations . As you go on in mathemat-ics you will solve many more quadratic equations.You can use the quadratic formula but it is hardto remember reliably. An alternative is the tech-nique of completing the square . It is based onthe binomial formulas which you should have usedso often in this class that you can’t help rememberingthem. If in fact you do not remember them, look themup and do so many exercises involving them that youwon’t be able to forget them if you tried.

Occasionally it is obvious how a quadratic expression or polynomialcan be factored and if so you look for when the in-dividual factors are zero.The equation x2 � 4x � 2 � 0 has two real solutions.Enter the smaller here and the larger here

4.(10 pts) Here is another quadratic equation. Ithas a conjugate complex pair of solutions.The solution of the equation x2 � 8x � 20 � 0 is1 i

5.(10 pts) Quadratic Equations do notalways occur in standard form. . Whenthey occur in disguise you apply the fun-damental principle of equation solving:figure out what bothers you and get rid of it by doing the same thing on both sides of the equation. If you understand and fully appreciate that principleyou have grasped more than half of Math 1010!Sometimes the process of converting an equation toa quadratic equation introduces spurious solutions.They solve the quadratic equation but not the originalequation. That makes it doubly important that youcheck your answers. You check not just to make sureyou didn’t make a mistake, but also to make sure youidentify only the solutions of the original equation.

Even if you can guess the answer to the equa-tion below, figure out how to solve it properly any-way, and check the solutions for two alternative ap-proaches when the set closes.

The equation

x � 2�

x � 3 � 0

has the real solutions x � .6.(10 pts) Inequalities are solved like equations,

by figuring out what bothers us and getting rid of itby doing the same thing on both sides of the inequal-ity. The only difference is that when multiplying witha negative factor we reverse the inequality. Thus wereplace � with � , * with + , etc.The solutions of the inequality

5 � 2x3� 7

2

satisfyx .(Enter ” � ” or ” � ” and a rational number.)

7.(10 pts) Polynomials are functions or expres-sions that can be evaluated in a finite number of ad-ditions, multiplications, and subtractions. However,they require no division for their evaluation. (Frac-tions are considered constants in this context). Poly-nomials have a whole language associated with themthat you need to understand. You also need to be ableto manipulate polynomial expressions to obtain theirstandard form .Consider the polynomial

p � x � � x2 � 1 �� x � 2 � 3x � 1 �Its degree is and its leading coefficient is.In standard form it can be written asp � x � x3 � x2 � x � .Note that some of these answers may be negative.

8.(10 pts) Rational Functions of Expressions areratios (or quotients) of polynomials . They are pro-cessed exactly like fractions .

Polynomials in standard form are sums of pow-ers of the variable. Numbers in decimal represen-tation are sums of powers of 10. Polynomials areprocessed much like numbers, except that you cannottrade some powers of x for a higher power of x. Thusthe manipulation of polynomials is in some ways ac-tually easier than the corresponding manipulation ofdecimal numbers!

An example for this fact is provided bylong division .Use long division with remainder to complete theblanks below. Remember that some of the coeffi-cients may be negative.2x3 � 2x2 � 5x � 5 � � x2 � x � 2 � � x � �

x � .

9.(10 pts) Perhaps the most central concept in all ofmathematics is that of a function . You need to un-derstand the concepts of rule , domain , and range ,and what it means to evaluate a function at a numberor an algebraic expression that may itself be definedby a function.

For this and the next two problems let

f � x � x � 1x2 � 5x � 6

�Two numbers not in the domain of f are and

. (Enter the numbers in increasing size.)

10.(10 pts) Consider the function from the previ-ous problem and evaluatef � � 2 � .

11.(10 pts) Again, consider the function in the pre-vious problem and evaluatef � x � 1 � .

12.(10 pts) The Number System is built in stagesto facilitate the four basic operations. It is extendedto complex numbers to facilitate the computation ofsquare roots of negative real numbers. All you needto remember to work with complex numbers is thatthey work exactly like algebraic expressions of theform a � bi where a and b are real numbers, exceptthat

i2 �� 1 �There is a trick related to dividing complex numbers:you multiply numerator and denominator with theconjugate complex of the denominator. This makesthe denominator real, and the standard from of thecomplex number follows from the distributive law .

Express the complex fraction below in standardform. Remember that the real or the imaginary partmay be negative.1 � 3i2 � i � � i.

13.(10 pts) Linear systems of equations arisefrequently in applications. They are usually solvedby Gaussian Elimination and Backward Substitu-tion. When working by hand it is crucial to orga-nize the computation in a clear way that lets youcheck your calculations so far. A linear systemmay have none, one, or infinitely many solutions.Solve the linear system

4x � y � 32x � y � 5

x � and y � .

3

14.(10 pts) Solve the linear system

x � 2y � 3z � 7� x � 2y � z � 1� x � y � 2z � � 2

x � , y � , and z � .15.(10 pts) A major tool in elementary math-

ematics is the graphing of equations in therectangular (or Cartesian) coordinate system .The interplay between algebra and geometry can beexploited to gain insights that would otherwise bemuch hard to obtain. Particularly important are equa-tions whose graphs are straight lines .The line that passes through � 1 � 3 and has slope 2 canbe written in general form asy � x - � 0.

16.(10 pts) The compute a distance between two pointswe simply apply the Pythagorean Theorem

The distance between the points � 2 � � 3 and � 1 � 1)is

17.(10 pts) Powers and radicals provide a beauti-ful illustration of how mathematics starts from simplebeginnings and then expands in a consistent manneruntil it reaches results and concepts (like anythingnon=zero to the power zero equals one) that aren’tobvious and may seem strange at first.

If 32x � 18 , then x � .

18.(10 pts) The key to manipulatingrational functions and expressions is that theywork exactly like fractions .

The rational function

r � x � x � 1x � 1

� 2x � 3x � 2

can be written in standard form asr � x � � �� .

19.(10 pts) There are some basic procedures forsolving word problems such as use common sense,draw a picture, introduce a variable, set up and solvean equation, have expectations, check your answers.Word problems are unpopular with students sincethey add a layer of complexity, remember that you arestudying mathematics precisely because it enablesyou to solve word problems.

The length of a rectangle is 4 inches more than itswidth. Its area is 96 square inches. The length of therectangle is inches and its width is inches.

20.(10 pts) It takes you 8 hours to dig a hole. Itwould take you and your brother 5 hours to dig thatsame hole together. If you brother was to dig the holeby himself it would take him hours.

21.(0 pts) Dear Students,We would very much appreciate your help in evaluating WeBWorK. We would like to hear about both thestrengths and weaknesses of WeBWorK as you see them and any suggestions that you have for improving theprogram or the way it is used. WeBWorK has already benefited from past comments by students and we wantto continue to improve it and make it as useful as possible as an educational tool.All questions are optional, but we appreciate all the information you give us. Thank you very much for yourtime.

Please select the best choice for each question.

A. For which mathematics course are you filling out this questionnaire?

A. � Math 1010B.� Math 1030C.� Math 1050D.� Math 1060E.� Math 1070

4

F.� Math 1090G.� Math 1100H.� Math 1210I.� Math 1220J.� Math 2210

K.� Other (please specify)

B. What is your gender?A. � Female

B.� Male

C. Is this your first time taking this course?A. � Yes

B.� No

D. What was the most recent previous math course, if any, that you were enrolled in at this university?A. � Math 1010

B.� Math 1030C.� Math 1050D.� Math 1060E.� Math 1070F.� Math 1090

G.� Math 1100H.� Math 1210I.� Math 1220J.� none

K.5

� Other (please specify)

E. What was the content of your most recent high school math course?A. � Calculus

B.� Pre-CalculusC.� AlgebraD.� GeometryE.� Other (please specify)

F. What is your ethnicity?A. � Asian American/Pacific Islander

B.� Black/African AmericanC.� Caucasian/WhiteD.� Latino/Hispanic AmericanE.� Native American/Alaskan NativeF.� Other (please specify)

G. What is your academic status?A. � Freshman

B.� SophomoreC.� JuniorD.� SeniorE.� Other (please specify)

H. What is your intended major?A. � Social Sciences (i.e. Psychology, Ecomonics, Political Science, History, etc.)

B.� Humanities (i.e. English, Religion and Classics, Languages, etc.)6

C.� Natural Sciences (i.e. Physics, Chemistry, Biology, etc.)D.� EngineeringE.� MathematicsF.� Other (please specify)

I. Do you plan to enroll in other mathematics courses at this university?A. � Yes

B.� No

J. What grade do you expect to receive for this class?A. � A

B.� BC.� CD.� DE.� E

K. Approximately how many days before weekly due dates do you typically begin to work on WeBWorKassignments?

A. � 1B.� 2C.� 3D.� 4E.� Other (please specify)

L. How many hours per week do you typically spend on WeBWorK problem sets?A. � Less than 1 hour

B.� 1 hourC.� 2 hours

7

D.� 3 hoursE.� More than 3 hours

M. How would you rank yourself as a mathematics student?

A. � Among the bestB.� Above averageC.� AverageD.� Below averageE.� Among the worst

N. Have you used WeBWorK in courses previous to this one?

A. � YesB.� No

O. Do you have access to a computer in your dorm room or residence?

A. � YesB.� No

P. Where do you typically work on WeBWorK problem sets?

A. � Your dorm roomB.� Your off campus residenceC.� A campus computing facilityD.� The mathematics departmentE.� The libraryF.� Other (please specify)

Please rate the frequency with which you do each of the following:¡TABLE BORDER CELLSPACING=1 CELLPADDING=4 WIDTH=600¿ ¡TR¿¡TD COLSPAN=”2” VALIGN=”TOP”

¿Scale: 5 all the time, 4 almost all the time, 3 sometimes, 2 almost never, 1 never, 0 no answer¡/TD¿ ¡/TR¿¡TR¿¡TD WIDTH= ”55

8

� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”451. Use the email connection to get help with specific problems ¡/TD¿¡/TR¿¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5

¡/TD¿ ¡TD WIDTH=”452. Do WeBWorK assignments with other students ¡/TD¿ ¡/TR¿ ¡TR¿¡TD WIDTH=”55 � 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”453. Get an entire assignment correct ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”454. Seek help from your teaching assistant ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”455. Seek help from your instructor ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 39

� 4� 5¡/TD¿ ¡TD WIDTH= ”456. Guess at problems you don’t understand ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”457. Get frustrated with and give up on a particular problem due to mathematicaldifficulty ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”458. Get frustrated with the time it takes WeBWorK to respond to answers you submitto it ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”459. Get frustrated with the syntactic requirements of answers you submit to WeBWorK¡/TD¿ ¡/TR¿

¡/TABLE¿Please rate the extent to which you agree with each of the following statements:¡TABLE BORDER CELLSPACING=1 CELLPADDING=4 WIDTH=600¿ ¡TR¿¡TD COLSPAN=”2” VALIGN=”TOP”

¿ 5 strongly agree, 4 agree, 3 neutral, 2 disagree, 1 strongly disagree, 0 no answer¡/TD¿ ¡/TR¿¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4510. I prefer WeBWorK over paper and pencil homework ¡/TD¿¡/TR¿¡TR¿¡TD WIDTH= ”55� 0

10

� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH=”4511. WeBWorK problems are challenging ¡/TD¿ ¡/TR¿ ¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4512. Class lectures effectively prepare me to complete WeBWorK assignments ¡/TD¿¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4513. The content of WeBWorK problems is consistent with the material taught inlectures ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4514. The content of WeBWorK problems is consistent with the material tested onexams ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4515. WeBWorK effectively prepares me for course examinations ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 211

� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4516. The immediate responses I get from WeBWorK make me more persistent withassignments ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4517. The immediate responses I get from WeBWorK help me learn the course material¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4518. Email access to professors is a useful component of WeBWorK ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4519. The Email feedback mechanism has made it easy to communicate with myprofessor ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4520. I can successfully access WeBWorK whenever I need to ¡/TD¿ ¡/TR¿ ¡TR¿¡TDWIDTH= ”55� 0� 1� 2� 3

12

� 4� 5¡/TD¿ ¡TD WIDTH= ”4521. I know where to go to get help when I am having trouble with course material orWeBWorK problems ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4522. WeBWorK makes mathematics courses more enjoyable ¡/TD¿ ¡/TR¿ ¡/TABLE¿

Please rate your satisfaction with each of the following:¡TABLE BORDER CELLSPACING=1 CELLPADDING=4 WIDTH=600¿ ¡TR¿¡TD COLSPAN=”2” VALIGN=”TOP”

¿ 5 highly satisfied, 4 satisfied, 3 neutral, 2 dissatisfied, 1 strongly dissatisfied, 0 no answer¡/TD¿ ¡/TR¿¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4523. Your course instructor ¡/TD¿¡/TR¿¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5

¡/TD¿ ¡TD WIDTH=”4524. Your course TA ¡/TD¿ ¡/TR¿ ¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 4� 5¡/TD¿ ¡TD WIDTH= ”4525. Your course ¡/TD¿ ¡/TR¿

¡TR¿¡TD WIDTH= ”55� 0� 1� 2� 3� 413

� 5¡/TD¿ ¡TD WIDTH= ”4526. WeBWorK ¡/TD¿ ¡/TR¿

¡/TABLE¿27. Please tell us what you like about WeBWorK.

28. Please tell us what you do not like about WeBWorK

29. Please use this space for additional comments regarding WeBWorK

Thank you very much.


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Tom Robbins MATH 1010-90 Fall 2002Final Exam, Questions, Answers, and Solutions

These are the questions on the final exam, for your information. You can answer them to test your under-standing, and you can see the solutions and answers.

1.(0 pts)Simplify45 � 2

312 � 3

7� /


6x � 3 � 12 � 3 � x � 1 has the solution x �

3.(0 pts)The equation x2 � 2x � 1 � 0 has two real solutions.Enter the smaller here and the larger here

4.(0 pts)One solution of the equation x2 � 2x � 3 � 0 is� i


1x � 2

� 1x � 3

� 1 � 0

has two real solutions. Enter the smaller hereand the larger here

6.(0 pts)The solutions of the inequality

4 � 3x2� 6

satisfyx .(Enter ” � ” or ” � ” and a rational number.)

7.(0 pts)Consider the polynomial

p � x � � x2 � 1 �� x � 2 � 3x � 4 �Its degree is and its leading coefficient is .In standard form it can be written asp � x � x3 + � x2 � x �

8.(0 pts)Use long division with remainder to complete theblanks below. Remember that some of the coeffi-cients may be negative.

5x3 � x2 � 3x � 1 � � x2 � 2x � 1 � � x � �x � .

9.(0 pts) For this and the next two problems let

f � x � x � 4x2 � 3x

�Two numbers not in the domain of f are and

. (Enter the numbers in increasing size.)10.(0 pts) Consider the function from the previous

problem and evaluate f � 5 � .11.(0 pts) Again, consider the function in the pre-

vious problem and evaluate f � 2x � 1 �.

12.(0 pts) Express the complex fraction below instandard form. Remember that the real or the imagi-nary part may be negative.2 � 3i1 � 4i � � i.

13.(0 pts) Solve the linear system

4x � y � 22x � y � 4

x � and y � .14.(0 pts) Solve the linear system

x � y � z � 6x � 2y � z � 2

2x � 3y � z � 5

x � , y � , and z � .15.(0 pts) The line that passes through � 1 � 1 and

has slope � 1 can be written in general form asy � x - � 0.

16.(0 pts) The distance between the point � � 1 � 2 and the origin is

17.(0 pts) If 125x � 25, then x � .18.(0 pts) The rational function

r � x � 1x � 1

� 1x � 2

� 2x � 3

can be written in standard form asr � x � � �� .

1

19.(0 pts) You have a backyard that’s three timesas long as it is wide. Within one yard of its boundaryyou plant shrubs and flowers, and the remainder youcover with 340 square yards of lawn. The width ofyour lawn is yards and its length is yards.

20.(0 pts) It takes you 6 hours to dig a hole. Ittakes your brother 8 hours to dig the same hole. Youryounger sister takes 12 hours. It takes the three ofyou hours to dig that hole.


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2

Math 1010-90 online

Answers for Exam 3

Tom RobbinsThis set contains no questions, just the answers to

Exam 3. You can look at the individual problems, ordownload a hard copy with all the answers as usual.Let me know if you have any questions!

1.(0 pts) A quadratic equation. Solve the equa-tion

x2 � 2x � 4 � 0 �As you know, I recommend that you solve a qua-

dratic equation by completing the square:

x2 � 2x � 4 � 0 � � 5x2 � 2x � 1 � 5 � perfect square� x � 1 2 � 5 � �

x � 1 � 1 � 5 � � 1x � 1 1 � 5 � the answer

2.(0 pts) Another quadratic equation. Solve theequation

4x2 � 12x � 5 � 0 �Dividing by the leading coefficient and completing

the square gives:

4x2 � 12x � 5 � 0 � � 4

x2 � 3x � 54 � 0 � � 1

x2 � 3x � 94 � 1 � perfect square:

x � 32 ; 2 � 1 � �

x � 32 � 1 1 � � 3

2

x � 32 1 1 � the answer

Thus x � 52 or x � 1

2 .

3.(0 pts) Yet another quadratic equation.Solvethe equation

x � 3�

x � 2 � 0by converting it to a quadratic equation.

Think of�

x as the unknown, and call it z. Thenthe above equation turns into

z2 � 3z � 2 � 0

which can be solved as usual. It can also be factoredto give

z2 � 3z � 2 � � z � 1 � z � 2 � 0

Thus z � 1 or z � 2. This corresponds to x � 1 orx � 4. Both solve the original equation.

4.(0 pts) A Radical Equation.Solve the equation�x � � x � 1 � 2

We are bothered by the square roots and we get ridof them by isolating and squaring one at a time. Thus:�

x � � x � 1 � 2 � � � x � 1�x � 2 � � x � 1 �<�= 2x � 4 � 4

�x � 1 � x � 1 � � 5 � x� 5 � � 4

�x � 1 �<�= 2

25 � 16 � x � 1 � � 169 � 16x � � 16x � 9

16 � the answer

Thus there is one solution, i.e., x � 916 .

5.(0 pts) Evaluating polynomials. Let

p � x � 2x2 � x � 3 �Then

p � x � 2 � 2 � x � 2 2 � � x � 2 � 3� 2:x2 � 4x � 4 ; � x � 2 � 3� 2x2 � 8x � 8 � x � 5� 2x2 � 9x � 13

6.(0 pts) Synthetic Division. Compute the ratio ofx3 � 4x2 � 7x � 6 and x � 2.

We proceed as described here :

2 : 1 � 4 7 � 62 � 4 6

1 � 2 3 01

Thusx3 � 4x2 � 7x � 6

x � 2 � x2 � 2x � 3 �7.(0 pts) Graphs of functions. Draw the graph of

the quadratic function

f � x � � x � 1 2 � 1 �What is the vertex of this parabola?

f is given in the completed square form a � x � h 2 �k, and the vertex � h � k = � � 1 � 1 of the parabola canbe read off it. Since a � 1 the parabola opens upward.

Its graph is shown in this Figure:

8.(0 pts)Rational Expressions. Write

1 � 2x

1 � 1x

in the standard form of a rational expressions (i.e., asthe ratio of two polynomials).

The easiest way to handle this expression is to mul-tiply with x in the numerator and denominator, giv-ing:

1 � 2x

1 � 1x� x � 2

x � 1�

9.(0 pts) Linear Systems. Solve the linear system

2x � y � 8x � 2y � 1

Subtracting twice the second equation from thefirst gives 3y � 6 and hence y � 2. Substituting y inthe first equation gives 2x � 2 � 8 and hence x � 5.

10.(0 pts)A word problem. The other day I drove home

150 miles from Southern Utah. It took me three hoursalong Interstate 15. I was driving at 75 miles per houruntil I reached Provo. Then I got stuck in traffic anddrove at 30 miles (on average) per hour until I reachedmy home. How far is my home from Provo? Hint:denote the distance from Southern Utah to Provo byx and the distance from Provo to my home by y, set upa linear system, and solve it. Time is distance dividedby speed.

The time spent traveling at 30 mph is x30 , and the

time spent traveling at 75 mph is 150 � x75 . Proceed-

ing as described in the hint we have the equationsx � y � 150 and x

75� y

30 � 3 � It follows from the firstequation that x � 150 � y. Substituting x in the sec-ond equation gives

150 � y75

� y30 � 3 �

This is a single linear equation in y which can besolved as follows:

150 � y75� y

30 � 3 � � 150300 � 2y � 5y � 450 � � 300

3y � 150 � � 3y � 50 � the answer

My home is 50 miles from Provo. Indeed, travel-ing 100 miles at 75 mph takes an hour and 20 min-utes, and traveling 50 miles at 30 mph takes an hourand 40 minutes, for a total of 3 hours.


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