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Example 1: Factoring
2, x, start superscript, 2, end superscript, plus, 8, x, plus, 3, x, plus, 12
First, notice that there is no factor common to all terms in
2, x, start superscript, 2, end superscript, plus, 8, x, plus, 3, x, plus, 12
. However, if we group the first two terms together and the last two terms together, each group has its own GCF, or greatest common factor:
In particular, there is a GCF of 2, x in the first grouping and a GCF of 3 in the second grouping. We can factor these out to obtain the following expression:
Trinomials of the Form x^2 + bx + c
To factor polynomials of the form x 2 + bx
+ c, begin with two pairs of parentheses with x at the left of
each.
( x)( x)
Next, find two integers whose product is c and whose
sum is b and place them at the right of the parentheses.
Example 1
Factor x 2 + 8 x + 12.
x 2 + 8 x + 12 = ( x)(
x)
12 can be factored in a variety of ways:
(–3)(–4) (3)(4), (–2)(–6), (2)(6), (–1)(–12), (1)(12),
Only one of those pairs of factors sum to 8, namely (2)(6),
so
x 2 + 8 x + 12 = ( x
+ 2)( x + 6)
Example 2
Factor x 2 – 7 x – 18.
–18 can be factored in the following ways:
(–3)(6) (3)(–6), (–2)(9), (2)(–9), (–1)(18), (1)(–18),
The only combination whose sum is also –7 is (2)(–9), so
x 2 – 7 x + 18 = ( x
+ 2)( x – 9)
Example 3
Factor x 2 – 6 x + 9.
9 can be factored as
(–3)(–3) (3)(3), (–1)(–9), (1)(9),
The only combination whose sum is –6 is (–3)(–3), so
x 2 – 6 x + 9 = ( x
– 3)( x – 3) = ( x – 3) 2
