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

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