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Simplifying

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Multiplying and Dividing

Rational Expressions

Remember that a rational numbercan be expressed as a quotient oftwo integers. A rational expression

can be expressed as a quotient oftwo polynomials.

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Simplify:7 x 7

x

2

1Step 1: Factor the numerator and

the denominator completely lookingfor common factors.

7 x 7 7( x

1) x 2 1 ( x 1)( x 1)

Next

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7 x 7 x 2 1

7( x 1)( x 1)( x 1)

What is the common factor? x 1

Step 2: Divide the numerator anddenominator by the common factor.

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7( x 1)

( x 1)( x 1)

7( x 1)

( x 1)( x 1)

1

1

Step 3: Multiply to get your answer.

Answer :7

x 1

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Looking at the answer from the previous example, what value of x would make the denominator 0?

x= -1

The expression is undefined when thevalues make the denominator equal to 0

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How do I find the values

that make an expressionundefined?

Completely factor the originaldenominator.

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Ex :2ab (a 2)(b 3)

3ab (a 2 4)

3ab (a 2 4) 3ab (a 2)(a 2)

The expression is undefined when:a= 0, 2, and -2 and b= 0.

Factor the denominator

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Lets go through another example

3a 3 a 4

2 a 3 6a 2

3a 3 a 4

2 a 3 6a 2

a 3 (3 a )2 a 2 ( a 3)

Factor

out theGCF

Next

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3

22 ( 3)

(3 )a

a a

a 3 factored is 1( 3)a a

cancel like factors3

2

1 ( 3)2 ( 3)

a a

a a

1

1

3

2

1( 3)2 ( 3)

a

a a

a

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3

2

1

2

a

a

2cancel out the like factor a

12

a

1

a

answer

What values is the original expression undefined?

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Now try to do some on your own.

2

2

3 2

3 2

5 6

1) 95 10

2)6 16

x x

x

x x

x x x

Also find the values that make

each expression undefined?

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Remember how to multiply fractions:

First you multiply the numeratorsthen multiply the denominators.

5 2:

6 20

Ex 10 1

120 12

5 2

6 20

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The same method can be used to

multiply rational expressions.

Ex :4 a 2

5ab 3 3bc

12 a 3 4 a a 3 b c

5 a b b b 12 a a a

11 1 1 1

1 1 1 1

c5b2 a 2

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Lets do another one.

Ex : x 3 3 x 2

x 2

5 x

6

x 2 10 x 9

x 2

6 x 27Step #1: Factor the numerator andthe denominator.

x 2 ( x 3)( x 6)( x 1)

( x 1)( x 9)( x 9)( x 3)

Next

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Step #2: Divide the numerator anddenominator by the common factors.

x 2 ( x 3)( x 6)( x 1)

( x 1)( x 9)( x 9)( x 3)

1

1

1

1

1

1

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Step #3: Multiply the numeratorand the denominator.

x 2

x 6

Remember how to divide fractions?

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Multiply by the reciprocal of thedivisor.

45

1625

45

2516

4 255 16

1

1

5

4

54

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Dividing rational expressions uses thesame procedure.

Ex: Simplify

y 2 y2 10 y 24

y2

2 y y2 2 y 8

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y 2 y2 10 y 24

y2 2 y y2 2 y 8

y 2 y2 10 y 24

y2 2 y 8 y2 2 y

y 2( y 12)( y 2)

( y 4)( y 2) y ( y 2)

1

1

1 1

Next

4( 12) y

y y

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Now you try to simplify theexpression:

x 3 x 2 4 x 12

2 x 2 6 x x 2

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Answer :1

2 x ( x 6)

Now try these on your own.

1)x + 3

2x 3 2 x 2

x 2 7 x 6 x 2 10 x 21

2)3 x 6

7 x

7

5 x 10

14 x

14

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Here are the answers:

1) x 6

2 x 2

( x 7)

2) 6( x 1)

5( x 1)