Algebra Rules Review Sheet Page 4
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4
REVIEW OF ALGEBRA
2
x
16
EXAMPLE 7
Simplify
.
2
x
2x
8
SOLUTION
Factoring numerator and denominator, we have
2
x
16
x
4 x
4
x
4
2
x
2x
8
x
4 x
2
x
2
To factor polynomials of degree 3 or more, we sometimes use the following fact.
The Factor Theorem
If
P
is a polynomial and
P b
0
, then
x
b
is a factor
6
of
P x
.
3
2
EXAMPLE 8
Factor
x
3x
10x
24
.
3
2
SOLUTION
Let
P x
x
3x
10x
24
. If
P b
0
, where is an integer, then is
b
b
a factor of 24. Thus, the possibilities for are
b
1,
2,
3,
4,
6,
8,
12,
and
24
.
We find that
P 1
12
,
P
1
30
,
P 2
0
. By the Factor Theorem,
x
2
is a
factor. Instead of substituting further, we use long division as follows:
2
x
x
12
3
2
x
2 x
3x
10x
24
3
2
x
2 x
2
x
10x
2
x
2x
12x
24
12x
24
3
2
2
Therefore
x
3x
10x
24
x
2 x
x
12
x
2 x
3 x
4
COMPLETING THE SQUARE
Completing the square is a useful technique for graphing parabolas or integrating rational
2
functions. Completing the square means rewriting a quadratic
ax
bx
c
2
in the form
a x
p
q
and can be accomplished by:
1.
Factoring the number from the terms involving .
a
x
2.
Adding and subtracting the square of half the coefficient of .
x
In general, we have
b
2
2
ax
bx
c
a x
x
c
a
2
2
b
b
b
2
a x
x
c
a
2a
2a
2
2
b
b
a x
c
2a
4a
2
EXAMPLE 9
Rewrite
x
x
1
by completing the square.
1
SOLUTION
The square of half the coefficient of is . Thus
x
4
(
)
2
1
1
1
3
2
2
x
x
1
x
x
1
x
4
4
2
4
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