Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 Section 5.5 Dividing Polynomials Copyright...

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

Section 5.5

DividingPolynomials



Objective #1 Use the quotient rule for exponents.


When dividing exponential expressions with the same nonzero base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent on the common base.

0, bbb

b nmn

m

The Quotient Rule


Objective #1: Example

1a. Divide the expression using the quotient rule: 12

45

5

12

12 44

8

55

5

5



1a. Divide the expression using the quotient rule: 12

45

5

12

12 44

8

55

5

5



1b. Divide the expression using the quotient rule: 9

2x

x

9

9 22

7

xx

x

x


Objective #2 Use the zero-exponent rule.


If b is any real number other than 0,

0 1.b

The Zero Exponent Rule


Only 2 is raised to the 0 power.

2xy is raised to the 0 power.

Only y is raised to the 0 power.

5 is raised to the 0 power.

Zero as an Exponent

12

22

1)2(

15

0

0

0

0

xxy

xy

EXAMPLESEXAMPLES

Simplify:



2a. Use the zero-exponent rule to simplify the expression: 014

The zero-exponent rule states that if b is any real number other than 0, then 0 1.b Thus, 014 1 .



2b. Use the zero-exponent rule to simplify the expression: 010

Note that only the 10 is raised to the 0 power.

0 010 1 10

1 1

1



2c. Use the zero-exponent rule to simplify the expression: 020x

Note that only the x is raised to the 0 power.

020 20 1

20

x


Objective #3 Use the quotients-to-powers rule.


If a and b are real numbers and b is nonzero, then

When a quotient is raised to a power, raise the numerator to the power and divide by the denominator raised to the power.

n

nn

b

a

b

a

Quotients to Powers Rule for Exponents


10

6

25

232

5

3

33

33

)(

)(

822

b

a

b

a

b

a

xxx

Cube the numerator and denominator.

Square the numerator and denominator.

Quotients-to-Powers Rule for Exponents

EXAMPLESEXAMPLES Simplify:


10

6

25

232

5

3

33

33

)(

)(

822

b

a

b

a

b

a

xxx


Square the numerator and denominator.

Quotients-to-Powers Rule for Exponents

EXAMPLESEXAMPLES Simplify:


6

12

32

343

32

34

3

2

4

125

)(

)(5

)(

)5(

5

y

x

y

x

y

x

y

x


Cube each factor in the numerator.

Simplify.

Quotients to Powers Rule

EXAMPLEEXAMPLE Simplify:



3a. Simplify the expression using the

quotients-to-powers rule: 2

5x

2 2

2

2

5 5

25

x x

x



3b. Simplify the expression using the

quotients-to-powers rule:

410

32a

b

410 4 10 4

3 3 4

40

12

2 2 ( )

( )

16

a a

b b

a

b


Objective #4 Divide monomials.


To divide monomials, divide the coefficients and then divide the variables. Use the quotient rule for exponents to divide the variable factors: Keep the variable and subtract the exponents.

Dividing Monomials


2

2

1

10

5 :Divide

4

37

3

7

x

x

x

x

Divide the coefficients, 5/10 = 1/2, then

divide the variables by subtracting exponents.

Simplify.

Dividing Monomials

EXAMPLEEXAMPLE


22

1335

3

35

3

3

2

6 :Divide

yx

yx

yx

yx

Divide the coefficients, 6/2 = 3, then divide the variables by subtracting exponents.

Simplify.

Dividing Monomials

EXAMPLEEXAMPLE



4a. Divide: 4

43

15

x

x

4

4 44

0

3 315151515

xx

x

x



4b. Divide: 6 5

29

3

x y

xy

6 5

6 1 5 22

5 3

9 933

3

x yx y

xy

x y


Objective #5 Check polynomial division.



5. Check your answer to 4b by showing that the product of the divisor and the quotient is the dividend.

2 5 3 1 5 2 3

6 5

(3 )(3 ) 3 3

9

xy x y x y

x y

6 55 3

29

33

x yx y

xyFrom 4b.


Objective #6 Divide a polynomial by a monomial.


Division of Polynomials

Dividing a Polynomial That Is Not a Monomial by a

MonomialTo divide a polynomial by a monomial, divide each term of the polynomial by the monomial.


Division of Polynomials

EXAMPLEEXAMPLE

Divide: .10502040 3322334 yxyxyxyx

SOLUTIONSOLUTION

33

22334

10

502040

yx

yxyxyx

33

2

33

23

33

34

10

50

10

20

10

40

yx

yx

yx

yx

yx

yx

2

524

xyyx

Express the division in a vertical format.

Divide each term of the polynomial by the monomial. Note the 3 separate quotients.

Simplify each quotient.


6 4 2

6 4 2

6 1 4 1 2 1

5 3

5 2 10Divide:

10

5 2 10

10 10 10

1 1

2 5 1

2 5

x x x

x

x x x

x x x

x x x

x xx

Divide the coefficients, then divide the variables by subtracting exponents.

Divide each term of the polynomial by the monomial.

Simplify.

Dividing a Polynomial by a Monomial



6a. Divide: 9 5 3

215 6 9

3

x x x

x

9 5 3 9 5 3

2 2 2 2

7 3

15 6 9 15 6 9

3 3 3 3

5 2 3

x x x x x x

x x x x

x x x



6b. Divide: 7 6 2 3 2

218 6 60

6

x y x y xy

xy

7 6 2 3 2 7 6 2 3 2

2 2 2 2

6 4

18 6 60 18 6 60

6 6 6 6

3 10

x y x y xy x y x y xy

xy xy xy xy

x y xy

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 Section 5.5 Dividing Polynomials Copyright...

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