copyright © 2011 pearson education, inc. polynomials chapter 5.1exponents and scientific notation...

Copyright © 2011 Pearson Education, Inc.

PolynomialsCHAPTER

5.1 Exponents and Scientific Notation5.2 Introduction to Polynomials5.3 Adding and Subtracting Polynomials5.4 Exponent Rules and Multiplying

Monomials5.5 Multiplying Polynomials; Special

Products5.6 Exponent Rules and Dividing

Polynomials

55


Exponents and Scientific Notation5.15.11. Evaluate exponential forms with integer exponents.2. Write scientific notation in standard form.3. Write standard form numbers in scientific notation.

3Copyright © 2011 Pearson Education, Inc.

Objective 1Evaluate exponential forms with integer exponents.


Evaluating Exponential Forms with Negative BasesIf the base of an exponential form is a negative number and the exponent is even, then the product is positive.If the base is a negative number and the exponent is odd, then the product is negative.

Raising a Quotient to a PowerIf a and b are real numbers, where b 0 and n is a natural number, then

.

n n

n

a ab b


Example 1Evaluate each exponential form.a. b. c.

Solution a.

b.

c.

56 43

56 6 6 6 6 6 7776

43 3 3 3 3 81

337

337

3

3

37

27343


Zero as an ExponentIf a is a real number and then

If a is a real number, where a 0 and n is a natural number, then

0,a 0 1.a

1 . nna

a


Example 2Rewrite with a positive exponent; then if the expression is numeric, evaluate it.a. b. c.

Solution a.

b.

c.

410 34

410 4

110

110,000

Rewrite using then simplify.1 , nna

a

3

14

34

14 4 4

1 164 64

22

22 2

12

12 2

14


continuedd. e. f.

Solution

d.

e.

f.

4x 35

3

15

35

24y

4x4

1x

24y 2

14y

2

4y

1125


If a is a real number, where a 0 and n is a natural number, then 1 . n

n aa

If a and b are real numbers, where a 0 and b 0

and n is a natural number, then

.

n na bb a


Example 3Rewrite with a positive exponent; then if the expression is numeric, evaluate it. a. b.

Solution a.

b.

3

13 3

1x

3

13

33 3 3 3 27

3

1x

3x


Example 4Rewrite with a positive exponent, then if the expression is numeric, evaluate it.a. b.

Solution

a.

b.

323

323

332

3

3

32

278

6mn

6mn

6nm

6

6

nm


Objective 2Write scientific notation in standard form.


Scientific notation: A number expressed in the form where a is a decimal number with and n is an integer.

Scientific notation gives us a shorthand way to write very large or very small numbers.

10 ,na1 10 a

Changing Scientific Notation (Positive Exponent) to Standard FormTo change from scientific notation with a positive integer exponent to standard form, move the decimal point to the right the number of places indicated by the exponent.


Example 5Write in standard form.

SolutionMultiplying 4.26 by 105 means that the decimal point will move five places to the right.

54.26 10

54.26 10 426,000


Changing Scientific Notation (Negative Exponent) to Standard FormTo change from scientific notation with a negative exponent in standard form, move the decimal point to the left the same number of places as the absolute value of the exponent.


Example 6Write in standard form.

SolutionMultiplying 3.87 by 10–4 is equivalent to dividing by 104, which causes the decimal point to move four places to the left.

43.87 10

43.87 10 4

3.8710

0.000387


Objective 3Write standard form numbers in scientific notation.


Changing Standard Form to Scientific NotationTo write a number greater than 1 in scientific notation:1. Move the decimal point so that the number is greater

than or equal to 1 but less than 10. (Tip: Place the decimal point to the right of the first nonzero digit.)

2. Write the decimal number multiplied by 10n, where n is the number of places between the new decimal position and the original decimal position.

3. Delete zeros to the right of the last nonzero digit.


Example 7Write 875,000 in scientific notation.Solution Place the decimal to the right of the first

nonzero digit, 8. Next, we count the places to the right of this decimal position, which is five places. We write the 5 as the exponent of 10. Finally, we delete all the 0s to the right of the last nonzero digit, which is 5 in this case.

875,000 58.75 10Move the decimal point here to express a number whose absolute value is greater than or equal to 1 but less than 10.

There are five places to the right of the new decimal point.


Changing Standard Form to Scientific NotationTo write a positive decimal number that is less than 1 in

scientific notation:1. Move the decimal point so that the number is greater

than or equal to 1 but less than 10. (Tip: Place the decimal point to the right of the first nonzero digit.)

2. Write the decimal number multiplied by 10n , where n is a negative integer whose absolute value is the number of places between the new decimal position and the original decimal position.

3. Delete zeros to the left of the first nonzero digit.


Example 8Write 0.0000000472 in scientific notation.Solution Place the decimal point between the 4 and 7

digits, so that we have 4.72, which is a decimal number greater than 1 but less than 10. Since there are eight decimal places between the original decimal position and the new position, the exponent is –8.

0.0000000472 Move the decimal point here to express a number whose absolute value is greater than or equal to 1 but less than 10.

8.4 72 10

There are eight decimal places in between the original position and the new position, so the exponent is –8.


Evaluate –2–6 .

a)

b)

c)

d) 64

12

164

164

5.1


Write 13,030,000 in scientific notation.

a)

b)

c)

d)

41303 10

613.03 10

71.303 10

90.1303 10

5.1


Write in standard notation.

a) 4,690,000

b) 4,690,000,000

c) 0.000000469

d) 0.00000469

64.69 10

5.1


Introduction to Polynomials5.25.21. Identify monomials.2. Identify the coefficient and degree of a monomial.3. Classify polynomials.4. Identify the degree of a polynomial.5. Evaluate polynomials.6. Write polynomials in descending order of degree.7. Combine like terms.


Objective 1Identify monomials.


Monomial: An expression that is a constant, a variable, or a product of a constant and variable(s) that are raised to whole number powers.


Example 1Is the given expression a monomial? Explain.a. 18 b. –0.4a2b c. 5a2 + 4b –

1Answera. 18 is a monomial because it is a constant.b. –0.4a2b is a monomial because it is a product of a

constant, –0.4, and variables, a2 and b, which have whole-number exponents.

c. 5a2 + 4b – 1 is not a monomial because it is not a product of a constant and variables. Instead, addition and subtraction are involved.


Objective 2Identify the coefficient and degree of a monomial.


Coefficient of a monomial: The numerical factor in a monomial.

Degree of a monomial: The sum of the exponents of all variables in a monomial.


Example 2Identify the coefficient and degree of each monomial.a. b. 9

Answer a. We can express as . In this form, we can see

that our coefficient is –2 and the exponents for the variables are 1 and 3. Since the degree is the sum of the variables’ exponents, the degree is 4.

b. Since 9 = 9x0 , where x is any real number except 0, we can see that 9 is the coefficient and 0 is the degree.

32 xy

32 xy 1 32 x y


Objective 3Classify polynomials.


Polynomial: A monomial or an expression that can be written as a sum of monomials.

Examples: 4x, 4x + 8, 2x2 + 5xy + 8y

Polynomial in one variable: A polynomial in which every variable term has the same variable.

Example: x2 – 5x + 2 is a polynomial in one variable

Binomial: A polynomial containing two terms.Trinomial: A polynomial containing three terms.

Degree of a polynomial: The greatest degree of any of the terms in the polynomial.


Example 3Determine whether the expression is a monomial, binomial, trinomial, or none of these. a. 4ab2 b. –9x2 + z c. 4n3+ 2n – 1Answer

a. 4ab2 is a monomial because it has a single term.

b. –9x2 + z is a binomial because it contains two terms.

c. 4n3+ 2n – 1 is a trinomial because it contains three terms.


continuedDetermine whether the expression is a monomial, binomial, trinomial, or none of these. d. x3 + 9x2 – x + 4 e. Answer

d. Although x3 + 9x2 – x + 4 is a polynomial, it has no special name because it has more than three terms.

e. is not a polynomial because is not a

monomial.

68yx

68yx

6x


Objective 4Identify the degree of a polynomial.


Example 4aIdentify the degree of the polynomial.

5 2 6 35 6 8x x x x x

Answer The degree is 6 because it is the greatest degree of all the terms.


Example 4bIdentify the degree of the polynomial.

4 5 2 28 2 9 1 a a b ab b

Answer To determine the degree of the monomial –2a5b2, add the exponents of its variables: 5 + 2 = 7Add the exponents of the variables in ab2: 1 + 2 = 3 Compare the degrees of all the terms: 4, 7, 3, 1 7 is the greatest degree. Therefore, 7 is the degree of the polynomial.


Objective 5Evaluate polynomials.


Example 5Evaluate when c = –6.2 4 7 c c

Solution 2 4 7 c c

2 + 46 6 7

36 24 7

19

Replace c with –6.

Simplify.


Example 6If we neglect air resistance, the polynomial −16t2 + h0 describes the height of a falling object from an initial height h0 for t seconds. If a rock is dropped from a building that is 120 feet tall, what is the height of the rock after it falls for 2 seconds?

Answer Evaluate −16t2 + h0 when t = 2 and h0 = 120.

−16(2)2 + 120 = 56= −64 + 120

After 2 seconds, the height of the rock is 56 feet.


Objective 6Write polynomials in descending order of degree.


Writing a Polynomial in Descending Order of DegreeTo write a polynomial in descending order of degree, place the highest degree term first, then the next highest degree, and so on.


Example 7Write the polynomial in descending order.

2 3 42 5 7 4 x x x x

Solution Rearrange the terms so that the highest degree term is first, then the next highest degree, and so on.

4 3 24 7 2 +5 x x x x

Degree 4 Degree 3 Degree 2 Degree 1 Degree 0

Answer 4 3 24 7 2 5 x x x x


Objective 7Combine like terms.


Example 8 Combine like terms and write the resulting polynomial in descending order of degree.

Solution

3 2 3 34 6 3 7 4 3 6x x x x x x

38 x 26x 2x 4

3 28 6 2 4x x x

3 2 3 34 6 3 7 4 3 6x x x x x x 3 3 3 24 3 6 4 6 7 3x x x x x x


Example 9 Combine like terms.

Solution

5 2 5 26 2 3 1 3 3 a a b b a b a b

5 2 5 26 2 3 1 3 3 a a b b a b a b5 5 2 23 2 3 3 6 1 a a a b a b b b

52 a 23 a b 0 5

5 22 +3 5a a b


continued Alternative Solution Instead of first collecting like terms, we strike through like terms in the given polynomial as they are combined.

5 2 5 26 2 3 1 3 3 a a b b a b a b

52 a 23 a b 0 5

5 22 +3 5a a b


Classify the expression

a) Monomial

b) Binomial

c) Trinomial

d) None of these

2 33 5.x y

5.2


Evaluate when x = –3.

a) –118

b) –10

c) 10

d) 134

3 23 5 8 x x

5.2


Identify the degree of the polynomial.

a) 3

b) 5

c) 6

d) 7

3 2 4 6 56 5 8 4 x x y x y y

5.2


Adding and Subtracting Polynomials5.35.31. Add polynomials.2. Subtract polynomials.


Objective 1Add polynomials.


We can add and subtract polynomials in the same way that we add and subtract numbers. In fact, polynomials are like whole numbers that are in an expanded form. In our base-ten number system, each place value is a power of 10. We can think of polynomials as a variable-base number system, where x2 is like the hundreds place (102) and x is like the tens place (101). To add whole numbers, we add the digits in like place values; in polynomials, we add like terms.

Adding PolynomialsTo add polynomials, combine like terms.


Example 1Add and write the resulting polynomial in descending

order of degree.

Solution

3 2 3 24 6 8 2 4 3 a a a a a a

3 2 3 24 6 8 2 4 3 a a a a a a

36 a3 26 +2 5 a a

22 a 0 5


Example 2Add and write the resulting polynomial in descending

order of degree.

Solution

4 2 4 22 57 2.5 3 3 4 73 6

a ab ab a ab ab

34 a

3 2 14 1.5 46

a ab ab

21.5ab16

ab 4

4 2 4 22 57 2.5 3 3 4 73 6

a ab ab a ab ab


Example 3Write an expression in simplest form for the perimeter of the rectangle shown.

5b + 2

9b – 10

Understand Perimeter means the total distance around the shape. Therefore, we need to

add the lengths of all the sides.Plan The lengths of the sides are represented by polynomials. Therefore, we add the polynomials to represent the perimeter.


continuedExecute Perimeter = Length + Width + Length + Width

9 10 b 5 2 b 9 10 b 5 2 b

9 5 9 5 10 2 10 2 b b b b

28 16 b

Answer The expression for the perimeter is .28 16b


continuedCheck To check:

1. Choose a value for b and evaluate the original expressions for length and width.2. Determine the corresponding numeric perimeter.3. Evaluate the perimeter expression using the same value for b and verify that we get the same numeric perimeter. Let’s choose b = 2.

Length: 9 10b 9 2 1018 10 8

Width: 5 2b 5 2 2

10 2 12

Perimeter

408 812 12+ + + =


continuedNow evaluate the perimeter expression where b = 2 and we should find that the result is 40.

Perimeter expression:28 16b

28 2 1656 16

40

This agrees with our calculation above.


Objective 2Subtract polynomials.


Subtracting PolynomialsTo subtract polynomials,1. Write the subtraction statement as an equivalent

addition statement.a. Change the operation symbol from a minus sign to a plus sign.b. Change the subtrahend (second polynomial) to its additive inverse. To get the additive inverse, we change the sign of each term in the polynomial.

2. Combine like terms.


Example 4

Subtract. 3 2 3 24 6 2 1 2 5 4 6x x x x x x

Solution

Change the minus sign to a plus sign.

3 2 3 24 6 2 1 2 5 4 6x x x x x x

Change all signs in the subtrahend.

3 22 2 5x x x

3 2 3 24 6 2 1 2 5 4 6x x x x x x


Example 5Subtract. 3 2 3 28 5 3 1 2 8 4 3 x x x x x x

Solution 3 2 3 28 5 3 1 2 8 4 3 x x x x x x

Change the minus sign to a plus sign.

3 2 3 28 5 3 1 2 8 4 3 x x x x x x

Change all signs in the subtrahend.

3 26 3 4x x x

Subtract.Subtract.


Add and write the polynomial in descending order.

a)

b)

c)

d)

3 2 3 25 8 4 3 6 x x x x3 25 5 2 x x

3 26 5 2 x x

3 24 5 2 x x

3 24 11 10 x x

5.3


Subtract and write the polynomial in descending order.

a)

b)

c)

d)

3 2 3 28 2 4 10 5 7 2 u u u u u3 218 7 7 6 u u u

3 218 7 7 2 u u u

3 22 5 5 2 u u u

3 22 3 7 2u u u

5.3


Exponent Rules and Multiplying Monomials5.45.4

1. Multiply monomials.2. Multiply numbers in scientific notation.3. Simplify a monomial raised to a power.


Objective 1Multiply monomials.


Consider 23 • 24, which is a product of exponential forms. To simplify 23 • 24, we could follow the order of operations and evaluate the exponential forms first, then multiply. 23 • 24 = 8 • 16 = 128

2 2 2 2 22 2

43 2 2

7 2

However, there is an alternative. We can write the result in exponential form by first writing 23 and 24 in their factored forms.

23 means three 2s.

24 means four 2s.

Since there are a total of seven 2s multiplied, we can express the product as 27.

Notice that 27 = 128.


Product Rule for ExponentsIf a is a real number and m and n are integers, then am • an = am+n.

Multiplying MonomialsTo multiply monomials:1. Multiply coefficients.2. Add the exponents of the like bases.3. Write any unlike variable bases unchanged in the

product.


Example 1Multiply 6 4 .x x

SolutionBecause the bases are the same, we can add the exponents and keep the same base.

6 4x x 6 4x 10x


Example 2Multiply.a. b. 4 35 2x x 3 2 2 42 5

3 6

a bc a b

Solution 4 35 2x x 4 35 2 xa.

710 x

b. 3 2 2 42 53 6

a bc a b 3 2 1 4 22 5 3 6

a b c1

3

5 5 259

a b c

Multiply the coefficients and add the exponents of the like bases.


Example 3Write an expression in simplest form for the volume of the box shown.

Understand We are given a box with side lengths that are monomial expressions.

3b

5bb

Plan The volume of a box is found by multiplying the length, width, and height.


continuedExecute V lwh

3 5V b b b315V b

Answer The expression for volume is 15b3.

Check Since is an identity, substitute 2 for b and solve.

33 5 15b b b b

33 5 15b b b b

33 2 5 2 2 15 2 6 10 2 15 8

120 120


Objective 2Multiply numbers in scientific notation.


We multiply numbers in scientific notation using the same procedure we used to multiply monomials.

Monomials:

3 6 3 6

9

4 2 4 2

8

a a a

a

Scientific notation:

3 6 3 6

9

4 10 2 10 4 2 10

8 10


Example 4aMultiply . Write the answer in scientific notation.

3 44.5 10 5.7 10

Solution Multiply 4.5 and 5.7, then add the exponents for base 10s.

3 44.5 10 5.7 10 3 44.5 5.7 10 725.65 10 82.565 10

Note: The product is not in scientific notation. We must move the decimal point one place to the left and account for this by increasing the exponent by one.


Example 4bMultiply . Write the answer in scientific notation.

3 66.2 10 3.1 10

Solution Multiply 6.2 and 3.1, then add the exponents for base 10s.

3 66.2 3.1 10 319.22 10 21.922 10

Note: The product is not in scientific notation. We must move the decimal point one place to the left and account for this by increasing the exponent by one.

3 66.2 10 3.1 10


Objective 3Simplify a monomial raised to a power.


A Power Raised to a PowerIf a is a real number and m and n are integers, then (am)n = amn.

Raising a Product to a PowerIf a and b are real numbers and n is an integer, then (ab)n = anbn.

Simplifying a Monomial Raised to a PowerTo simplify a monomial raised to a power,1. Evaluate the coefficient raised to that power.2. Multiply each variable’s exponent by the power.


Example 5 Simplify. a. b. c.

49 62

3c d

Solution

3 6 36 a

366a 320.8xy

a. 366a

18216a

49 62

3c d

49 4 6 42

3c d

36 241681

c d

3 1 3 2 30.8 x y

3 60.512x y


Example 6a Simplify. 32 76 8y y

Solution Since the order of operations is to simplify exponents before multiplying, we will simplify (8y7)3 first, then multiply the result by 6y2.

233072 yMultiply coefficients and add exponents of like variables.

Simplify.

32 76 8y y 2 3 7 36 8 y y

2 216 512 y y2+216 512 y


Example 6b Simplify. 22 40.3 2 1.5x xy x y

Solution We follow the order of operations and simplify the monomials raised to a power first, then multiply the monomials.

Multiply coefficients and add exponents of like variables.

Simplify.

2 21 2 4 2 1 20.3 2 1.5x xy x y 22 40.3 2 1.5x xy x y

2 8 20.09 2 2.25x xy x y2 1 8 1 20.09 2 2.25x y

11 30.405x y


Simplify.

a)

b)

c)

d)

7 36 3b b

2118b

1018b

219b

109b

5.4


Simplify.

a)

b)

c)

d)

42 34 5y y

920y

2020y

92500y

142500y

5.4


Multiplying Polynomials; Special Products5.55.5

1. Multiply a polynomial by a monomial.2. Multiply binomials.3. Multiply polynomials.4. Determine the product when given special polynomial

factors.


Objective 1Multiply a polynomial by a monomial.


Multiplying a Polynomial by a MonomialTo multiply a polynomial by a monomial, use the distributive property to multiply each term in the polynomial by the monomial.


Example 1Multiply.a. 22 6 2 1 p p p

Solution 22 6 2 1 p p pa.

2 2 2 2 2 16 pp pp p3 212 4 2 p p p

2p • 6p2

2p • 2p2p • –1


continuedb. 2 3 3 42 3 5 4 a b a b ab a bcSolution

2 3 3 42 3 5 4 a b a b ab a bcb.

2 3 2 3 2 4 2 2 3 2 2 5 2 4 a b a b a b ab a b a a b bc5 2 3 4 6 2 2 6 2 10 8 a b a b a b a b c

Note: When multiplying multivariable terms, it is helpful to multiply the coefficients first, then the variables in alphabetical order.

–2a2b • 3a3b

–2a2b • ab3

–2a2b • – 5a4 –2a2b • 4bc


Objective 2Multiply binomials.


Multiplying PolynomialsTo multiply two polynomials,1. Multiply every term in the second polynomial by

every term in the first polynomial.2. Combine like terms.


Example 2

Multiply. 7 3x x

Solution Multiply each term in x + 7 by each term in x + 3.

7 3x x

+7 3 x x 3x 7 x

7 • x

7 • 3

x • 3

x • x

2= 3 7 21 x x x 2 10 = 21x x


Example 3Multiply. 2 1 5 x x

Solution Multiply each term in x – 5 by each term in 2x + 1 (think FOIL).

2 1 5 x x

+1 52 x x 2 5 x 1 xFirst Outer Inner Last

1 • xInner

1 • (–5)Last

2x • (–5)Outer2x •

xFirst


continued

22 10 5x x x

22 9 5x x

22 9 5x x

2 2 5 1 1 5 x x x x

2 1 5 x x


Example 4Multiply. 3 5 2 7 x x

Solution Multiply each term in 2x + 7 by each term in 3x – 5 (think FOIL).

3x • 2x3x • 7

(–5) • 2x

3 5 2 7 x x

(–5) • 7

+ 5 7 3 2 x x 3 7 x 5 2 x

First Outer Inner Last


continued 3 2 3 7 5 2 5 7 x x x x

2 6 21 10 35 x x x

2 6 11 35 x x

26 11 35 x x

3 5 2 7 x x


Example 5Multiply. 4 2 2 3x x

Solution Multiply each term in 4x − 2 by each term in 2x – 3 (think FOIL).

4x • 2x4x • (–3)

(–2) • 2x

4 2 2 3x x

(–2) • (–3)

+ 2 3 4 2x x 4 3x 2 2 x First Outer Inner Last


continued

2 8 12 4 6x x x

2 8 16 6x x

28 16 6x x

4 2 2 3x x

+ 2 3 4 2x x 4 3x 2 2 x


The product of two binomials can be shown in terms of geometry.

355x

7xx2

Length • width = Sum of the areas of the four internal rectangles

7 5 x x 2 5 7 35 x x x2 12 35 x x Combine like terms.


Objective 3Multiply polynomials.


Example 6Multiply. 23 2 3 3 x x x

Solution Multiply each term in 2x2 + 3x +3 by each term in x – 3.

23 2 3 3 x x x

2 2 2 3 3 3 2 3 3 3 3x x x x x x x 3 2 2 2 3 3 6 9 9 x x x x x3 2 2 3 6 9x x x

(–3) • 2x2

(–3) • 3x(–3) • 3

x • 2x2

x • 3xx • 3


Objective 4Determine the product when given special polynomial factors.


Multiplying ConjugatesIf a and b are real numbers, variables, or expressions, then (a + b)(a – b) = a2 – b2.

Conjugates: Binomials that differ only in the sign separating the terms.

x + 9 and x – 9 2x + 3 and 2x – 3

–6x + 5 and –6x – 5


Example 7Multiply.a. (x + 5) (x – 5)

b.

Solution 5 5 x x 2 25 x

2 25 xUse (a + b)(a – b) = a2 – b2.

Simplify.

4 3 4 3 a a

Solution 4 3 4 3 a a 2 24 3 a

216 9 a

Use (a + b)(a – b) = a2 – b2.

Simplify.


Squaring a BinomialIf a and b are real numbers, variables, or expressions, then (a + b)2 = a2 + 2ab + b2

(a – b)2 = a2 – 2ab + b2


Example 8aMultiply.

Solution

2 24 2 4 7 7a a Use (a + b)2 = a2 + 2ab + b2.

Simplify.

24 7a

24 7a 216 56 49a a


Example 8bMultiply.

Solution

2 23 2 3 8 8 y y Use (a – b)2 = a2 – 2ab + b2.

Simplify.

23 8y

23 8y29 48 64 y y


Multiply.

a)

b)

c)

d)

2 5 3 y y

22 15 y y

22 2 15 y y

22 15 y y

22 11 15 y y

5.5


Multiply.

a)

b)

c)

d)

25 4x

225 40 8 x x

225 20 8 x x

225 20 16 x x

225 40 16 x x

5.5


Exponent Rules and Dividing Polynomials5.65.6

1. Divide exponential forms with the same base.2. Divide numbers in scientific notation.3. Divide monomials.4. Divide a polynomial by a monomial.5. Use long division to divide polynomials.6. Simplify expressions using rules of exponents.


Objective 1Divide exponential forms with the same base.


Quotient Rule for Exponents

If m and n are integers and a is a real number, where a 0, then .

mm n

n

a aa


Example 1Divide.

8

4

xx

Solution Because the exponential forms have the same base, we can subtract the exponents and keep the same base.

8 4x x 8 4 4 x x8

4

xx


Example 2Divide and write the result with a positive exponent.

6

2

yy

Solution6 ( 2) y6 2 y

4y

4

1

y

Subtract the exponents and keep the same base.

Rewrite the subtraction as addition.

Simplify.

Write with a positive exponent.

6

2

yy


Objective 2Divide numbers in scientific notation.


Example 3Divide and write the result in scientific notation.

7

3

2.34 103.6 10

Solution The decimal factors and powers of 10 can be separated into a product of two fractions, allowing separate division. 7

3

2.34 103.6 10

7

3

2.34 103.6 10

7 30.65 10 40.65 10

36.5 10

Note: This is not in scientific notation. We must move the decimal point one place to the right and account for this by decreasing the exponent by one.


Objective 3Divide monomials.


Dividing MonomialsTo divide monomials,1. Divide the coefficients or simplify them to fractions

in lowest terms.2. Use the quotient rule for the exponents with like

bases.3. Do not change unlike variable bases in the quotient.4. Write the final expression so that all exponents are

positive.


Example 4Divide.

3 6 2

4 3

1824

x y zx y

Solution3 6 2

4 3

1824

x y zx y

3 6 2

4 3

18 24 1

x y zx y

3 4 6 3 218 24

x y z

1 3 234

x y z3 23 1

4 1 1

y zx

3 234

y z

x


Objective 4Divide a polynomial by a monomial.


If a, b, and c are real numbers, variables, or expressions with c 0, then

Dividing a Polynomial by a Monomial

To divide a polynomial by a monomial, divide each term in the polynomial by the monomial.

.

a b a bc c c


Example 5Divide.

6 2 3

2

36 9 63 x y x y xx y

Solution6 2 3

2

36 9 63 x y x y xx y

6 2 3

2 2 2

36 9 63 3 3

x y x y xx y x y x y

14 1 0 212 3 xx y x y

y

4 212 3x y xxy

Divide each term in the polynomial by the monomial.


Objective 5Use long division to divide polynomials.


To divide a polynomial by a polynomial, we can use long division.

Divide: 15712

157

12 37 36

1

1

1

23

Divisor

Quotient

Remainder

Quotient 13

Divisor 12

Remainder 1

= Dividend=157

•++

•


Example 6Divide.

2 8 163

x xx

Solution Begin by dividing the first term in the dividend by the first term in the divisor: x2 + 8x + 16.

2

2

3 8 16

3

xx x x

x x

Change signs.

2

2

3 8 16

3

xx x x

x x

5 16x


continuedThe next term in the quotient is found by dividing the term 5x + 16 by x + 3 .

Change signs.

Answer 153

xx

2

2

3 8 16

3

xx x x

x x

5 16x

5

5 15x

The next term in the quotient is found by dividing the term 5x + 16 by x + 3 .

2

2

3 8 16

3

xx x x

x x

5 15x 5 16x

1

5


Dividing a Polynomial by a PolynomialTo divide a polynomial by a polynomial, use long division. If there is a remainder, write the result in the following form:

remainderquotientdivisor


Example 7Divide.

26 5 282 1

b b

b

Solution Begin by dividing the first term in the dividend by the first term in the divisor: 6b2 + 5b – 28.

2

2

3 2 1 6 5 28

6 3

bb b b

b b

2

2

3 2 1 6 5 28

6 3 8 28

bb b b

b bb

Change signs.


continuedDetermine the next part of the quotient by dividing 8b by 2b, which is 4, and repeat the multiplication and subtraction steps.

Change signs.

2

2

3 4 2 1 6 5 28

6 3 8 28 8 4

bb b b

b bbb

2

2

3 4 2 1 6 5 28

6 3 8 28 8 4 2

4

bb b b

b bbb

Answer 243 42 1

bb


Example 8Divide.

4 28 10 2 94 2

x x xx

Solution Use a place holder for the missing terms.

4 3 2

4 2 8 0 10 2 9x x x x x


continuedSolution 4 3 2

4 3

3 2

3 2

2

2

4 2 8 0 10 2 9

8 4

4 10

4 2

12 2

12 6 4 9 4 2

x x x x x

x x

x x

x x

x x

x xxx

11

3 2 2 3 1x x x

3 2 11 2 3 14 2

x x xx

Answer

Remember to change signs when subtracting.


Objective 6Simplify expressions using rules of exponents.


Exponents SummaryAssume that no denominators are 0, that a and b are real numbers, and that m and n are integers.

Zero as an exponent: a0 = 1, where a 0.00 is indeterminate.

Negative exponents:

Product rule for exponents:Quotient rule for exponents:Raising a power to a power:Raising a product to a power:Raising a quotient to a power:

1 1, ,n nn n

a aa a

m n m na a am

m nn

a aa

nm mna a

n n nab a b

n na bb a

n

n

na ab b


Example 9Simplify. Write all answers with positive exponents.a.

Solution

37

3

xx

37

3

xx

37 3 x

310 x10 3 x30x

30

1

x

a.


continued b.

32

54

6

3

m

m

Solution

32

54

6

3

m

m

3 2 3

5 4 5

63

mm

6

20

216243

mm

6 208 9

m

148 9

m

14

89m

Use the quotient rule for exponents.

Write with a positive exponent.


Simplify.

a)

b)

c)

d)

25

3

aa

16

1a

4

1a

16a

4a

5.6


Divide.

a)

b)

c)

d)

2 6 102

y yy

1842

yy

242

yy

652

yy

282

yy

5.6

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