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4 ■ REVIEW OF ALGEBRA

EXAMPLE 7 Simplify

x 2 ⫺ 16
.
x ⫺ 2x ⫺ 8
2

SOLUTION Factoring numerator and denominator, we have

x 2 ⫺ 16
共x ⫺ 4兲共x ⫹ 4兲
x⫹4


x ⫺ 2x ⫺ 8
共x ⫺ 4兲共x ⫹ 2兲
x⫹2
2

To factor polynomials of degree 3 or more, we sometimes use the following fact.
6 The Factor Theorem If P is a polynomial and P共b兲 苷 0, then x ⫺ b is a factor

of P共x兲.
EXAMPLE 8 Factor x 3 ⫺ 3x 2 ⫺ 10x ⫹ 24.
SOLUTION Let P共x兲 苷 x 3 ⫺ 3x 2 ⫺ 10x ⫹ 24. If P共b兲 苷 0, where b is an integer, then b is

a factor of 24. Thus, the possibilities for b are ⫾1, ⫾2, ⫾3, ⫾4, ⫾6, ⫾8, ⫾12, and ⫾24.
We find that P共1兲 苷 12, P共⫺1兲 苷 30, P共2兲 苷 0. By the Factor Theorem, x ⫺ 2 is a
factor. Instead of substituting further, we use long division as follows:
x 2 ⫺ x ⫺ 12
x ⫺ 2 兲 x 3 ⫺ 3x 2 ⫺ 10x ⫹ 24
x 3 ⫺ 2x 2
⫺x 2 ⫺ 10x
⫺x 2 ⫹ 2x
⫺ 12x ⫹ 24
⫺ 12x ⫹ 24
Therefore

x 3 ⫺ 3x 2 ⫺ 10x ⫹ 24 苷 共x ⫺ 2兲共x 2 ⫺ x ⫺ 12兲
苷 共x ⫺ 2兲共x ⫹ 3兲共x ⫺ 4兲

COMPLETING THE SQUARE

In general, we have





ax 2 ⫹ bx ⫹ c 苷 a x 2 ⫹
苷 a x2 ⫹
苷a x⫹



b
x ⫹c
a

冉 冊 冉 冊册
冊 冉 冊

b
x⫹
a
b
2a

b
2a

2

2

⫹ c⫺



2

b
2a

⫹c

b2
4a

EXAMPLE 9 Rewrite x 2 ⫹ x ⫹ 1 by completing the square.
1

SOLUTION The square of half the coefficient of x is 4. Thus
2

x 2 ⫹ x ⫹ 1 苷 x 2 ⫹ x ⫹ 14 ⫺ 14 ⫹ 1 苷 (x ⫹ 12 ) ⫹ 34

Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.

Completing the square is a useful technique for graphing parabolas or integrating rational
functions. Completing the square means rewriting a quadratic ax 2 ⫹ bx ⫹ c
in the form a共x ⫹ p兲2 ⫹ q and can be accomplished by:
1. Factoring the number a from the terms involving x.
2. Adding and subtracting the square of half the coefficient of x.