11.2 Simplify Rational Expressions

Rational ExpressionRational Expression - an expression that can

be written as a ratio of two polynomials where the denominator is not 0.

A rational expression is undefined when the denominator is 0.

Excluded value – number that makes a rational expression undefined.

Example: is undefined when . So, 3 is an excluded value

3

2

x3x

Example 1 Find excluded values

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

10x8+xa.

+ 14

5

2yb.

c.9

4v

v 2 – + 5

2+7w

w+8w 2

d.

ANSWER The excluded value is 0.

SOLUTION

The expression is undefined when or10x

8+x10x 0, =

x 0.=

a.

Example 1

The expression is undefined when

or x =

b.+ 14

5

2y+ 142y 0,=

7.–

ANSWER The excluded value is 7.–

ANSWER The excluded values is and 3– 3.

Find excluded values

c. The expression is undefined when or

The solutions of the equation are

3 and 3.

94v

v 2 –

0,= 9v 2 –

( )3–v3v +( ) 0.=

–

Example 1

ANSWER There are no excluded values.

The expression is undefined when

The discriminant is So, the quadratic equation has no real roots.

d.+ 52+7w

w+8w 2

+ 5w+8w 2 = 0.

4–b 2 1

2=4ac– ( )8 ( )5 0.<

Find excluded values

Simplifying a Rational Expression1. Factor the numerator & denominator.2. Divide out any common factors.

A rational expression is in simplest form if the numerator and denominator have no factors in common other than 1.

Example 2 Simplify expressions by dividing out monomials

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

a. 2rr

b. 5x( )2+x5

18m 2

12m 26m

3c. – d. y

y7 –

=21

Simplify.

SOLUTION

Divide out common factor.2rr

a. =2rr

ANSWER The excluded value is 0.

Simplify.=2+x

x

ANSWER The excluded value is 2.–

c.18m2

12m 26m

3 –=

6m 2( )2m –

m 236 • •

Factor numerator and denominator.

Example 2 Simplify expressions by dividing out monomials

=6m

2( )2m

m 236 • •

Divide out common factors.–

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

=5

( )2+x5

x

••

ANSWER The excluded value is 7.

Simplify.=2m –

3

d. y

y7 –The expression is already in simplest form.

Example 2 Simplify expressions by dividing out monomials

ANSWER The excluded value is 0.

Example 3

Simplify to lowest terms.

2(x + 5)

3x + 1

Multiple Choice Practice

1–9x 2

32x+6x 2 10+

2(x + 5)

3x – 1

2(3x + 5)

3x – 1

2(x + 1)

9(x – 1)

SOLUTION

1–9x 2

32x+6x 2 10+

=2(x + 5)

(3x – 1) (3x + 1)

(3x + 1) Factor numerator and denominator.

Example 3

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

(3x – 1) (3x + 1)

(3x + 1)

Simplify.=2(x + 5)

3x – 1

ANSWER The correct answer is B.

Multiple Choice Practice

Example 4 Recognize opposites

Simplify . State the excluded values.x

2–16

7x–x 2 12+

x 2–16

7x–x 2 12+

= (4 – x) (4 + x)

(x – 3) (x – 4) Factor numerator and denominator.

=– (x – 4) (4 + x)

(x – 3) (x – 4)Rewrite 4 – x as –(x – 4).

=– (x – 4) (4 + x)

(x – 3) (x – 4)Divide out common factor.

=– (4 + x)

x – 3Simplify.

Example 4 Recognize opposites

=x + 4

x – 3Write as .– a

b– ba

–

ANSWER The excluded values are – 4 and 4.

Example 5 Simplify a rational model

CELL PHONE COSTS

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 the rational function

18x

2.2x46C

–=

100 – + 2.2x 2

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

Example 5

SOLUTION

18x

2.2x46C

–=

100 – + 2.2x 2

Write model.

180x

22x460 –=

1000 – + 22x 2

Multiply numerator and denominator by 10.

90x

11x230 –=

500 – + 11x 2( )

( )

2

2 Factor numerator and denominator.

Simplify a rational model

Divide out common factor.90x

11x230=

500 – + 11x 2( )

( )

2

2 –

Example 5

Simplify.90x

11x230 –=

500 – + 11x 2

Simplify a rational model

11.2 Warm-Up