Simplifying

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 of two polynomials.

Remember, denominators cannot = 0.

Now, let’s 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

Simplify: 7x 7

x2 1

Step 1: Factor the numerator and the denominator completely looking for common factors:

Numerator:

Denominator:

7x 7 7(x 1)

x2 1 (x 1)(x 1)

7x 7

x2 1

7(x 1)

(x 1)(x 1)

What is the common factor? x 1

Step 2: Divide the numerator and denominator by the common factor.

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

Looking at the answer from the previous example …

what value of x would make the denominator 0?x = (-1) makes the denom=0

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

How do I find the values that make an expression undefined?

Completely factor the original denominator.

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

3ab(a2 4)

3ab(a2 4) 3ab(a 2)(a 2)

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

Factor the denominator

Lets go through another example.

3a3 a4

2a3 6a2

3a3 a4

2a3 6a2 a3 (3 a)

2a2 (a 3)

Factor out the GCF

Next

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

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?

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??

2

2

3 2

3 2

5 61)

9

5 102)

6 16

x x

x

x x

x x x

1) x² - 5x + 6 = (x – 2)(x – 3) x² - 9 (x + 3)(x – 3)

Cancel out both (x – 3)’s

= (x – 2) (x + 3)

2) 5x³ + 10x² = 5x²(x + 2)

x³ - 6x² - 16x x(x² - 6x – 16)

= 5x²(x + 2) x(x – 8)(x + 2)

Cancel out both (x + 2)’s

= 5x² = 5x x(x – 8) (x – 8)

Do you 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

The same method can be used to multiply rational expressions.

Ex: 4a2

5ab3 3bc

12a3 4 a a 3 bc

5 a b b b 12 a a a

11 1 1 1

1 1 1 1

c

5b2 a2

Let’s do another one.

Ex: x3 3x2

x2 5x 6

x2 10x 9

x2 6x 27Step #1: Factor the numerator and the denominator.

x2 (x 3)

(x 6)(x 1)(x 1)(x 9)

(x 9)(x 3)Next

Step #2: Divide the numerator and denominator by the common factors.

x2 (x 3)

(x 6)(x 1)(x 1)(x 9)

(x 9)(x 3)1

1

1

1

1

1

Step #3: Multiply the numerator and the denominator.

x2

x 6

Remember how to divide fractions?

Multiply by the reciprocal of the divisor.

4

5

16

25

4

525

16

4 25

516

1

1

5

4

5

4

Dividing rational expressions uses the same procedure.

Ex: Simplify

y 2

y2 10 y 24

y2 2y

y2 2y 8

y 2

y2 10 y 24

y2 2y

y2 2y 8

y 2

y2 10 y 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

Now you try to simplify the expression:

x 3

x2 4x 12

2x2 6x

x 2

Answer: 1

2x(x 6)

Now try these on your own.

1) x + 3

2x3 2x2

x2 7x 6

x2 10x 21

2) 3x 67x 7

5x 1014x 14

Here are the answers:

1) x 6

2x2 (x 7)

2) 6(x 1)5(x 1)

Now wasn’t that fun???!!!

Your homework is: