3.8 Simplifying Rational Expressions p. 161-164. Vocabulary Rational Expression: ratio of 2 polynomials Rational Expression: ratio of 2 polynomials Excluded.

3.8 Simplifying Rational 3.8 Simplifying Rational ExpressionsExpressions

p. 161-164p. 161-164

VocabularyVocabulary Rational Expression: ratio of 2 polynomialsRational Expression: ratio of 2 polynomials

Excluded Value: once factored (before Excluded Value: once factored (before simplified), the value that would make the simplified), the value that would make the denominator 0denominator 0

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

Excluded ValueExcluded Value Set denominator = 0 (you know it can’t = 0)Set denominator = 0 (you know it can’t = 0)

Solve for the variableSolve for the variable

That is your excluded value.That is your excluded value.

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.



b. 2y + 14

5

SOLUTION

The expression 5 is undefined when

2y + 14 = 0, or y = – 7.2y + 14

ANSWER

The excluded value is – 7.



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


x + 23x – 5

1.

ANSWER

The excluded value is .53

3.

ANSWER

The excluded value is and 4 .32

–

n – 62n2 – 5n – 12

2. 2mm2 – 4

ANSWER

The excluded value is 2, and 2 .–

To simplifyTo simplify Factor numerator and denominatorFactor numerator and denominator

Cancel any common factors (anything over Cancel any common factors (anything over itself = 1)itself = 1)

EXAMPLE 2 Simplify expressions by dividing out monomials

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

a. r2r

SOLUTION

Divide out common factor.a. r2r = r

2r

= 12 Simplify.

ANSWER


EXAMPLE 2


b. 5x5(x + 2)

SOLUTION

b. 5x5(x + 2) = 5 x

5 (x + 2) Divide out common factor.

Simplify.= x(x + 2)

ANSWER


Simplify expressions by dividing out monomials

EXAMPLE 2


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



EXAMPLE 2


SOLUTION

d. y7 – y

d. The expression y7 – y

is already in simplest form.

ANSWER




5. 4 a3

22a62

11a3ANSWER The excluded value is 0.

6. 2c c + 5

2c c + 5ANSWER The excluded value is – 5.

7. 2s2 + 8s3s +12

ANSWER 2s3


8. 8x8x3 + 16x2

ANSWER1

x2 + 2xThe excluded values are 0 and – 2.

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 3 Simplify an expression by dividing out binomials

CHECKCheck your simplification using a graphing calculator.

Graph y1x2 – 3x – 10x2 + 6x + 8 and y2= = x – 5

x + 4

The graphs coincide. So, the expressions are equivalent for all values of x other than the excluded values (– 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) (x – 4)(4 + x)=

ANSWERThe 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 + 2x2 + 7x + 10

(x + 1)(x + 5)

ANSWER The excluded values are – 2 and – 5.

10.y2 – 64

y2 – 16y + 64 ANSWER

(y + 8)(y – 8)

The excluded value is 8

11. 5 + 4z – z2

z2 – 3z – 10ANSWER

(z + 1)(z + 2)

– The excluded values are 5 and – 2.

EXAMPLE 5 Simplify a rational model

46 – 2.2xC =

100 – 18x + 2.2x2

where x is the number of years since 1991. Rewrite the model so that it has only whole number coefficients.Then simplify the model.

The average cost C (in dollars per minute) for cell phone service in the United States during the period 1991–2000 can be modeled by

CELL PHONE COSTS

EXAMPLE 5 Simplify a rational model

C =46 – 2.2x

100 – 18x + 2.2x2Write model.

= 460 – 22x1000 – 180x + 22x2

Multiply numerator and denominator by 10.

Factor numerator and denominator.

=2(230 – 11x)

2(500 – 90x + 11x2)

Divide out common factor.

=2(230 – 11x)

2(500 – 90x + 11x2)

SOLUTION

= 230 – 11x

500 – 90x + 11x2Simplify.


In Example 5, approximate the average cost per minute in 2000.

12.

ANSWERThe average cost per minute in 2000 is $.23/min.

3.8 Simplifying Rational Expressions p. 161-164. Vocabulary Rational Expression: ratio of 2 polynomials Rational Expression: ratio of 2 polynomials Excluded.

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