3.8 Warm Up3.8 Warm Up1. (18x1. (18x³ - 24x² + 12x) ∕ 6x³ - 24x² + 12x) ∕ 6x

2. (5x² + 7x – 6) ∕ (x + 2)2. (5x² + 7x – 6) ∕ (x + 2)

3. (9x² + 5x – 6) ∕ (x + 1)3. (9x² + 5x – 6) ∕ (x + 1)

3.8 Simplifying Rational 3.8 Simplifying Rational ExpressionsExpressions

VocabularyVocabularyRational ExpressionRational Expression: ratio of 2 : ratio of 2

polynomialspolynomials

Excluded ValueExcluded Value: once simplified, the : once simplified, the value that would make the value that would make the denominator 0denominator 0

Simplest FormSimplest Form: when numerator & : when numerator & denominator have no common factorsdenominator have no common factors

EXAMPLE 1 Find excluded values

Find the excluded values, if any, of the expression.

a. x + 810x

SOLUTION

a. The expression x + 8 is undefined when 10x = 0, or x = 0.

10x

ANSWER

The excluded value is 0.

EXAMPLE 1 Find excluded values

Find the excluded values, if any, of the expression.

b. 2y + 14

5

SOLUTION

The expression 5 is undefined when

2y + 14 = 0, or x = – 7.2y + 14

ANSWER

The excluded value is – 7.

EXAMPLE 1 Find excluded values

Find the excluded values, if any, of the expression.

c. v 2 – 9

4v

SOLUTION

c. The expression 4v is undefined when v2 – 9 = 0,

or (v + 3)(v – 3) = 0. The solutions of the equation are

– 3 and 3.

v2 – 9

The excluded values are – 3 and 3.

ANSWER

GUIDED PRACTICE for Example 1

Find the excluded values, if any, of the expression.

x + 23x – 5

1.

ANSWER

The excluded value is .53

3.

ANSWER

The excluded value is and 4 .32

–

n – 6

2n2 – 5n – 12

2. 2mm2 – 4

ANSWER

The excluded value is 2, and 2 .–

EXAMPLE 2 Simplify expressions by dividing out monomials

Simplify the rational expression, if possible. State the excluded values.a. r

2r

SOLUTION

Divide out common factor.a. r2r = r

2r

=12 Simplify.

ANSWER

The excluded value is 0.

EXAMPLE 2

Simplify the rational expression, if possible. State the excluded values.b. 5x

5(x + 2)

SOLUTION

b. 5x5(x + 2)

= 5 x5 (x + 2) Divide out common factor.

Simplify.= x(x + 2)

ANSWER

The excluded value is – 2.

Simplify expressions by dividing out monomials

EXAMPLE 2

Simplify the rational expression, if possible. State the excluded values.

SOLUTION

c. 6m3 – 12m2

18m2

c.18m2

6m3 – 12m2 =

6m2 (m – 2)6 3 m2

Factor numerator and denominator.

=6m2 (m – 2)6 3 m2 Divide out common factors.

= m – 2 3

Simplify.

ANSWER

The excluded value is 0.

Simplify expressions by dividing out monomials

EXAMPLE 2

Simplify the rational expression, if possible. State the excluded values.

SOLUTION

d. y7 – y

d. The expression y

7 – yis already in simplest form.

ANSWER

The excluded value is 7.

Simplify expressions by dividing out monomials

GUIDED PRACTICE for Example 2

5. 4 a3

22a6

6.2c

c + 5

7. 2s2 + 8s3s +12

8.8x

8x3 + 16x2

EXAMPLE 3 Simplify an expression by dividing out binomials

Simplify x2 – 3x – 10x2 + 6x + 8

.State the excluded values.

SOLUTION

x2 – 3x – 10x2 + 6x + 8

=(x – 5)(x + 2)(x + 4)(x + 2)

Factor numerator and denominator.

(x – 5)(x + 2)(x + 4)(x + 2)

=

= x – 5x + 4

Divide out common factor.

Simplify.

ANSWER

The excluded values are – 4 and – 2.

EXAMPLE 4 Recognize opposites

Simplify x2 – 7x + 1216 – x2

.State the excluded values.

SOLUTION

x2 – 7x + 12 16 – x2

Factor numerator and denominator.

(x – 3)(x – 4)– (x – 4)(4 + x)

= Rewrite 4 – x as – ( x – 4).

Simplify.

– (x – 4)(4+ x)(x – 3)(x – 4)

= Divide out common factor.

–(4 + x)(x – 3)

=

(x – 3)(x – 4) (4 – x)(4 + x)=

ANSWER

The excluded values are – 4 and 4.

(x + 4)(x – 3)

= –

GUIDED PRACTICE for Examples 3 and 4

Simplify the rational expression. State the excluded values.

9. x2 + 3x + 2

x2 + 7x + 10

10.y2 – 64

y2 – 16y + 64

11.5 + 4z – z2

z2 – 3z – 10