Algebra Rules Review Sheet Page 3
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REVIEW OF ALGEBRA
3
x
x
y
1
2
y
y
x
y
x
x x
y
x
xy
(d)
2
y
x
y
y
x
y
y x
y
xy
y
1
x
x
FACTORING
We have used the Distributive Law to expand certain algebraic expressions. We sometimes
need to reverse this process (again using the Distributive Law) by factoring an expression
as a product of simpler ones. The easiest situation occurs when the expression has a com-
mon factor as follows:
Expanding
3x(x-2)=3x@-6x
Factoring
2
To factor a quadratic of the form
x
bx
c
we note that
2
x
r x
s
x
r
s x
rs
so we need to choose numbers
r and s
so that
r
s
b
and
rs
c
.
2
EXAMPLE 4
Factor
x
5x
24
.
SOLUTION
The two integers that add to give and multiply to give
5
24
are
3
and .
8
Therefore
2
x
5x
24
x
3 x
8
2
EXAMPLE 5
Factor
2x
7x
4
.
2
SOLUTION
Even though the coefficient of
x
is not , we can still look for factors of the
1
form
2x
r
and
x
s
, where
rs
4
. Experimentation reveals that
2
2x
7x
4
2x
1 x
4
Some special quadratics can be factored by using Equations 1 or 2 (from right to left)
or by using the formula for a difference of squares:
2
2
a
b
a
b a
b
3
The analogous formula for a difference of cubes is
3
3
2
2
4
a
b
a
b a
ab
b
which you can verify by expanding the right side. For a sum of cubes we have
3
3
2
2
a
b
a
b a
ab
b
5
EXAMPLE 6
2
2
(a)
x
6x
9
x
3
(Equation 2;
a
x, b
3
)
2
(b)
4x
25
2x
5 2x
5
(Equation 3;
a
2x, b
5
)
3
2
(c)
x
8
x
2 x
2x
4
(Equation 5;
a
x, b
2
)
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