chapter 6 rational expressions and equations section 6.1 multiplying rational expressions
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Chapter 6Rational Expressions and Equations
Section 6.1
Multiplying Rational Expressions
HW #6.1Pg 248 1-37Odd, 40-43
Chapter 6Rational Expressions and Equations
Section 6.2
Addition and Subtraction
9.
11. 12.
13. 14.
15. 16.
10.
9. 10. 11. 12.
13. 14. 15. 16.
LOGICAL REASONING Tell whether the statement is always true, sometimes true, or never true. Explain your reasoning.
1. The LCD of two rational expressions is the product of the denominators.• Sometimes
2. The LCD of two rational expressions will have a degree greater than or equal to that of the denominator with the higher degree.• Always
Simplify the expression.
17.
18.
19.
20.
HW #6.2 Pg 253-254 3-30 Every Third
Problem 31-45 Odd
Chapter 6Rational Expressions and Equations
6.3
Complex Rational Expressions
HW 6.3Pg 258 1-23 Odd, 26-28
HW Quiz 6.3Tuesday, April 18, 2023
( ) ( ) 1Evaluate for ( )
f x h f xf x
h x
Chapter 6Rational Expressions and Equations
6.4
Division of Polynomials
• Do a few examples of a poly divided by a monomial
• Discuss the proof of the remainder theorem
HW #6.4Pg 262 1-25 Odd, 26-32
Chapter 6Rational Expressions and Equations
Section 6.5
Synthetic Division
Dividing using Synthetic DivisionPart 1
Objective: Use synthetic division to find the quotient of certain polynomials
• Algorithm– A systematic procedure for doing certain computations.
• The Division Algorithm used in section 6.4 can be shortened if the divisor is a linear polynomial– Synthetic Division
Dividing using Synthetic DivisionPart 1
EXAMPLE 1
To see how synthetic division works, we will use long division to divide the polynomial by 3 22 3x x 3x
Dividing Polynomials
Using Synthetic Division
List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0.
1Set divisor = 0 and solve. Put answer here.
x + 3 = 0 so x = - 3
Synthetic Division
There is a shortcut for long division as long as the divisor is x – k where k is some number. (Can't have any powers on x).
3
286 23
x
xxx
- 3 1 6 8 -2
1
Bring first number down below lineMultiply these and
put answer above line
in next column
- 3 Add these up
3Multiply these and
put answer above line
in next column
- 9 Add these up
- 1
3
1
Multiply these and
put answer above line
in next column
Add these up
This is the remainder
Put variables back in (one x was divided out in process so first number is one less power than original problem).
x2 + x
So the answer is:
3
1132
xxx
List all coefficients (numbers in front of x's) and the constant along the top. Don't forget the 0's for missing terms.
1Set divisor = 0 and solve. Put answer here.
x - 4 = 0 so x = 4
Let's try another Synthetic Division
4
64 24
x
xx
4 1 0 - 4 0 6
1
Bring first number down below lineMultiply these and
put answer above line
in next column
4 Add these up
4Multiply these and
put answer above line
in next column
16 Add these up
12
48
48
Multiply these and
put answer above line
in next column
Add these up
This is the remainder
Now put variables back in (remember one x was divided out in process so first number is one less power than original problem so x3).
x3 + x2 + x +
So the answer is:
4
19848124 23
xxxx
0 x3 0 x
Multiply these and
put answer above line
in next column
192
198
Add these up
List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0.
You want to divide the factor into the polynomial so set divisor = 0 and solve for first number.
Let's try a problem where we factor the polynomial completely given one of its factors.
502584 23 xxx
- 2 4 8 -25 -50
4
Bring first number down below lineMultiply these and
put answer above line
in next column
- 8 Add these up
0Multiply these and
put answer above line
in next column
0 Add these up
- 25
50
0
Multiply these and
put answer above line
in next column
Add these up
No remainder so x + 2 IS a factor because it
divided in evenlyPut variables back in (one x was divided out in process so first number is one less power than original problem).
x2 + x
So the answer is the divisor times the quotient:
2542 2 xx
2 :factor x
You could check this by multiplying them out and
getting original polynomial
HW #6.5Pg 265 1-19
. . . And WhyTo solve problems using rational equations
6-6 Solving Rational Equation
A rational equation is an equation that contains one or more rational expressions. These are rational equations.
To solve a rational equation, we multiply both sides by the LCD to clear fractions.
Multiplying by the LCD
Multiplying to remove parentheses
Simplifying
2x =
3
120x = -
11
The LCD is x - 5, We multiply by x - 5 to clear fractions
5 is not a solution of the original equation because it results in division by 0, Since 5 is the only possible solution, the equation has no solution.
y = 57 No Solution
The LCD is x - 2. We multiply by x - 2.
The number -2 is a solution, but 2 is not since it results in division by O.
The solutions are 2 and 3.
e. x = 3 f. x = -3, 4 g. x = 1, -½ h. x = 1, -½
This checks in the original equation, so the solution is 7.
x = 7 x = -13
HW #6.6Pg 269 1-25 Odd, 26-34
Warm Up
Solve the following equation
6-7
Tom knows that he can mow a golf course in 4 hours. He also knows that Perry takes 5 hours to mow the same course. Tom must complete the job in 2! hours. Can he and Perry get the job done in time? How long will it take them to complete the job together?
If Perry gets a larger mower so that he can mow the course alone in 3 hours, how long will it take Tom and Perry to complete the job together?
Solving Work Problems
If a job can be done in t hours, then 1/t of it can be done in one hour. This is also true for
any measure of time.
Tom can mow a lawn in 4 hours. Perry can mow the same lawn in 5 hours. How long would it take both of them, working together with two lawn mowers, to mow the lawn?
Objective: Solve work problems using rational equations.
UNDERSTAND the problem
Question: How long will it take the two of them to mow the lawn together?
Tom can do 1/4 of the job in one hour
Perry can do 1/5 of the job in one hour
Data: Tom takes 4 hours to mow the lawn. Perry takes 5 hours to mow the lawn.
Tom can mow a lawn in 4 hours. Perry can mow the same lawn in 5 hours. How long would it take both of them, working together with two lawn mowers, to mow the lawn?
Objective: Solve work problems using rational equations.
Develop and carryout a PLAN
Let t represent the total number of hours it takes them working together. Then they can mow 1/t of it in 1 hour.
Translate to an equation.
1 1 14 5 t
Tom can do 1/4 of the job in one hour
Perry can do 1/5 of the job in one hour
Together they can do 1/t of the job in one hour
Tom can mow a lawn in 4 hours. Perry can mow the same lawn in 5 hours. How long would it take both of them, working together with two lawn mowers, to mow the lawn?
Objective: Solve work problems using rational equations.
Tom knows that he can mow a golf course in 4 hours. He also knows that Perry takes 5 hours to mow the same course. Tom must complete the job in 2! hours. Can he and Perry get the job done in time? How long will it take them to complete the job together?
If Perry gets a larger mower so that he can mow the course alone in 3 hours, how long will it take Tom and Perry to complete the job together?
1 1 14 3 t
5t = 1 hours
12
22
5t hours
Objective: Solve work problems using rational equations.
At a factory, smokestack A pollutes the air twice as fast as smokestack B.When the stacks operate together, they yield a certain amount of pollution in 15 hours. Find the time it would take each to yield that same amount of pollution operating alone.
1/x is the fraction of the pollution produced by A in 1 hour.
1/2x is the fraction of the pollution produced by B in 1 hour.
1/15 is the fraction of the total pollution produced by A and B in 1 hour.
1 1 1+ =
x 2x 15
Objective: Solve work problems using rational equations.
32 96A hours,B hours
An airplane flies 1062 km with the wind. In the same amount of time it can fly 738 km against the wind. The speed of the plane in still air is 200 km/h. Find the speed of the wind.
Objective: Solve motion problems using rational equations.
r = 36 km/h
Objective: Solve motion problems using rational equations.
Try This
d. A boat travels 246 mi downstream in the same time it takes to travel 180 mi upstream. The speed of the current in the stream is 5.5 mi/h. Find the speed of the boat in still water.
a. 35.5 mi/h
e. Susan Chen plans to run a 12.2 mile course in 2 hours. For the first 8.4 miles she plans to run at a slower pace, then she plans to speed up by 2 mi/h for the rest of the course. What is the slower pace that Susan will need to maintain in order to achieve this goal?
e. about 5.5 mi/h
Try This
Jorge Martinez is making a business trip by car. After driving half the total distance, he finds he has averaged only 20 mi/h, because of numerous traffic tie-ups. What must be his average speed for the second half of the trip if he is to average 40 mi/h for the entire trip? Answer this question using the following method.
1. Let d represent the distance Jorge has traveled so far, and let r represent his average speed for the remainder of the trip. Write a rational function, in terms of d and r, that gives the total time Jorge’s trip will take.
Try This
Jorge Martinez is making a business trip by car. After driving half the total distance, he finds he has averaged only 20 mi/h, because of numerous traffic tie-ups. What must be his average speed for the second half of the trip if he is to average 40 mi/h for the entire trip? Answer this question using the following method.
2. Write a rational expression, in terms of d and r, that gives his average speed for the entire trip.
Try This
Jorge Martinez is making a business trip by car. After driving half the total distance, he finds he has averaged only 20 mi/h, because of numerous traffic tie-ups. What must be his average speed for the second half of the trip if he is to average 40 mi/h for the entire trip? Answer this question using the following method.
3. Using the expression you wrote in part (b), write an equation expressing the fact that his average speed for the entire trip is 40 mi/h. Solve this equation for r if you can. If you cannot, explain why not.
HW #6.7 Pg 273 1-27 Odd, 29-33
6-8
PVT
K
We solve the formula for the unknown resistance r2.
We solve the formula for the unknown resistance r2.
HW #6.8Pg 278 1-30
What you will learn
1. Find the constant and an equation of variation for direct and joint variation problems.
2. To find the constant and an equation of variation for inverse variation problems
3. To solve direct, joint, and inverse variation problems
6-9
Objective: Find the constant of variation and an equation of variation for direct variation problems.
Direct Variation
Whenever a situation translates to a linear function f(x) = kx, or y = kx, where k is a
nonzero constant, we say that there is direct variation, or that y varies directly with x. The
number k is the Constant of Variation
Objective: Find the constant of variation and an equation of variation for direct variation problems.
The constant of variation is 16.
The equation of variation is y = 16x.
Objective: Find the constant of variation and an equation of variation for direct variation problems.
Objective: Find the constant of variation and an equation of variation for joint variation problems.
Joint Variation
y varies jointly as x and z if there is some number k such that y = kxz,
wherek 0, x 0, and z 0.
Objective: Find the constant of variation and an equation of variation for joint variation problems.
Suppose y varies jointly as x and z. Find the constant of variation and y when x = 8 and z = 3, if y = 16 when z = 2 and x = 5.
EXAMPLE 2
Find k
y = kxz
16 = k(2)(5)
16 810 5
k
85
y xz
88 3
5y
1925
y
Objective: Find the constant of variation and an equation of variation for joint variation problems.
Try This
Objective: Find the constant of variation and an equation of variation for inverse variation problems.
Inverse Variation
y varies inversely as x if there is some number k such that y = k/x,
wherek 0 and x 0.
EXAMPLE 3
Objective: Find the constant of variation and an equation of variation for inverse variation problems.
EXAMPLE 3
Objective: Find the constant of variation and an equation of variation for inverse variation problems.
Objective: Find the constant of variation and an equation of variation for inverse variation problems.
Try This
1
30
Describe the variational relationship between x and z and demonstrate this relationship algebraically.1. x varies directly with y, and y varies inversely
with z.
2. x varies inversely with y, and y varies inversely with z.
3. x varies jointly with y and w, and y varies directly with z, while w varies inversely with z.
The weight of an object on a planet varies directly with the planet’s mass and inversely with the square of the planet's radius. If all planets had the same density, the mass of the planet would vary directly with its volume, which equals 343
r1. Use this information to find how the weight
of an object w varies with the radius of the planet, assuming that all planets have the same density.
2. Earth has a radius of 6378 km, while Mercury (whose density is almost the same as Earth’s) has a radius of 4878 km. If you weigh 125 lb on Earth, how much would you weigh on Mercury?
HW #6.9 Pg 283-284 1-32
Chapter 6
Review
Two Parts
Part 1• Add/Subtract/Multiply/Divide Rational Expressions
• Solve Rational Equations
• Long Division/Synthetic Division
• Direct/Joint/Inverse Variation
• Challenge Problems
Part 2• Work Problems
• Distance Problems
• Problems with no numbers
• Challenge Problems
6 5 4
4
32 64 244
x x xx
Simplify
Simplify
Simplify
Simplify
Simplify
Simplify
Simplify
Simplify
7 11
xx
Solve
Solve
Divide
Divide
Compute the value of f x h f x
h
for 1
2f x
x
Find the value of k if (x + 2) is a factor of 3 2 5 6x kx x
HW # R-6 Pg 287-288 1-29
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