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Simplifying, Multiplying and Dividing Rational Expressions

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Simplifying Rational Expressions Let a, b, and c be nonzero real

numbers (or variable expressions). Then,

cb

ca

b

a

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Example: Simplify

Factor the numerator and denominator

Cancel common factors Rewrite the simplified expression

1

652

2

x

xx

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Example: Simplify1

652

2

x

xx

1

652

2

x

xx

)1)(1(

)1)(6(

xx

xx

1

6

x

x

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Multiplying Rational Expressions with Monomials Simplify as much as possible Multiply numerators Multiply denominators Simplify Rewrite the simplified expression

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Example: 2

43

22

32

18

10

2

6

y

yx

yx

yx

2

43

22

32

18

10

2

6

y

yx

yx

yx

36

60 33yx

3

5 33yx

2 5

3

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Multiplying Rational Expressions with polynomials Factor the numerators and

denominators of each rational expression

Simplify by dividing out common factors

Multiply Rewrite in simplified form (no

parentheses or negative exponents)

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Example:x

xx

xx

xx

3

143

123

273 2

2

3

x

xx

xx

xx

3

)1)(13(

)1)(13(

)91(3 2

x

xx

xx

xxx

3

)1)(13(

)1)(13(

)31)(31(3

)13()31( xx

29 6 1x x

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Dividing Rational Expressions Rewrite as a multiplication problem

(Multiply the first by the reciprocal of the second.)

Factor numerators and denominators Simplify Multiply Rewrite the answers in simplified form

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Example:6

3

84

32

2

xx

xx

x

6

3

84

32

2

xx

xx

x )2)(3(

)3(

)2(4

3

xx

xx

x

)3(

)2)(3(

)2(4

3

xx

xx

x x4

3

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Multiplying or Dividing Rational Expressions by Polynomials Treat the denominator of the

polynomial expression as 1. Factor each numerator and

denominator Multiply/”Divide” Rewrite in simplified form

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Example xxx

xx

22

2

44

3108

1

4

4

3108 2

2

2 xx

x

xx

)4(

1

4

310822

2

xxx

xx

)14(

1

4

)32)(14(2

xxx

xxxx

x

24

1)32(34

32

x

x

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Assignment

Page 558 #16-49