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

CHECK

In the graphing calculator activity on page 560, you saw how to use a graph to check a sum or difference of polynomials.

Check your simplification using a graphing calculator.

Graph y1x2 – 3x – 10x2 + 6x + 8

and y2= = x – 5x + 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)=

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

(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.