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:

(1)(12), (–1)(–12), (2)(6), (–2)(–6), (3)(4), (–3)(–4)

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:

(1)(–18), (–1)(18), (2)(–9), (–2)(9), (3)(–6), (–3)(6)

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

(1)(9), (–1)(–9), (3)(3), (–3)(–3)

The only combination whose sum is –6 is (–3)(–3), so

x 2 – 6 x + 9 = ( x – 3)( x – 3) = ( x – 3) 2