# Fichier PDF

Partage, hébergement, conversion et archivage facile de documents au format PDF

## 6e reviewofalgebra.pdf

Page 1 2 3 45623

#### Aperçu texte

4 ■ REVIEW OF ALGEBRA

EXAMPLE 7 Simplify

x 2 ⫺ 16
.
x ⫺ 2x ⫺ 8
2

SOLUTION Factoring numerator and denominator, we have

x 2 ⫺ 16

x⫹4

x ⫺ 2x ⫺ 8

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

COMPLETING THE SQUARE

In general, we have

ax 2 ⫹ bx ⫹ c 苷 a x 2 ⫹

b
x ⫹c
a

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