Factorizing Algebraic Expressions Worksheet Page 3
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182
2
Notice that in each of these examples, we end up with a quantity in the form A
. In
B
example 1, we have
2
2
2
= x
9
A
B
= (x + 3)(x
3)
where we have identified A = x and B = 3. In example 2, we have
2
2
2
A
B
= x
81
= (x + 9)(x
9)
where we have identified A = x and B = 9. The result that we have developed and have used
in two examples is called the difference of two squares, and is written:
2
2
= (A + B)(A
A
B
B)
The next common factorization that is important is called a perfect square. Notice that
2
(x + 5)
= (x + 5)(x + 5)
= x(x + 5) + 5(x + 5)
2
= x
+ 5x + 5x + 25
2
= x
+ 10x + 25
2
2
= x
+ 2(5x) + 5
The perfect square is written as:
2
2
2
(x + a)
= x
+ 2ax + a
Similarly,
2
(x
= (x
a)
a)(x
a)
= x(x
a)
a(x
a)
2
2
= x
ax + a
ax
2
2
= x
2ax + a
For example,
2
(x
7)
= (x
7)(x
7)
= x(x
7)
7(x
7)
2
2
= x
7x
7x + 7
2
= x
14x + 49
Page 3
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