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Simplifying

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Multiplying and Dividing

Rational Expressions

Remember that a rational number

can be expressed as a quotient of

two integers. A rational expression

can be expressed as a quotient oftwo polynomials.

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Remember, denominators

can not = 0.

Now,lets go through the steps to

simplify a rational expression.

Examples of rational expressions

2

4 8 4 7, ,

3 3 5 9

x y

x x y y

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Simplify:7x 7

x

2

1

Step 1: Factor the numerator and

the denominator completely lookingfor common factors.

Next

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7x 7 7(x1)

x2 1 (x1)(x1)

7x 7

x2 1

7(x1)

(x1)(x1)

What is the common factor?

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x1

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

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7(x1)

(x 1)(x1)

7(x1)

(x 1)(x1)

1

1

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Answer: 7

x1

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Step 3: Multiply to get youranswer.

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Looking at the answer from the

previous example, what value ofx

would make the denominator 0?

x= -1

The expression is undefined when the

values 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(a2

4)

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Factor the denominator

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3ab(a2

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

The expression is undefined when:

a= 0, 2, and -2 and b= 0.

ANSWER:

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Lets go through another example.

3a3 a4

2a3 6a2

3a3 a4

2a3

6a2

a3 (3 a)

2a2(a 3)

Factor

out the

GCF

Next

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3

2

2 ( 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

1

2

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 61)

9

5 102)

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 numerators

then 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:

4a2

5ab3

3bc

12a3 4 a a 3 b c

5 a b b b 12 a a a

11 1 1 1

1 1 1 1

c

5b2a2

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Lets do another one.

Ex:x3 3x2

x

2

5x

6

x2 10x 9

x

2

6x

27

Next

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Step #1: Factor the numerator

and the denominator.

x2

(x 3)(x 6)(x1)

(x1)(x 9)(x 9)(x3)

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Step #2: Divide the numerator and

denominator by the common factors.

x2 (x 3)

(x 6)(x1)

(x1)(x 9)

(x 9)(x3)

1

1

1

1

1

1

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Step #3: Multiply the numerator

and the denominator.x

2

x 6

Remember how to divide fractions?

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Multiply by the reciprocal of the

divisor. 45

1625

45

2516

4 25516

1

1

5

4

54

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Dividing rational expressions uses

the same procedure.

Ex: Simplify

y

2y

210y 24

y

2

2yy

2 2y 8

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y 2

y2

10y 24

y2 2y

y2

2y 8

y 2

y2

10y 24y2 2y 8

y2

2y

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 the

expression:

x 3

x2 4x12

2x2 6x

x 2

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Answer:

1

2x(x 6)

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Now its you time!

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1)x + 3

2x3 2x2

x2 7x 6

x2 10x 21

2) 3x 67x 7

5x 1014x 14

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THESE ARE THE ANSWERS!!

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1)x 6

2x2

(x 7)

2) 6(x 1)

5( x 1)