Factoring x2 + bx + c
We now begin a study of the factoring of trinomials. We first factor trinomials like
x2+ 5x + 6 and x2 + 3x – 10
by a refined trial-and-error process. In this section, we restrict our attention to trinomials of the type ax2 + bx + c where a = 1. The coefficient a is often called the leading coefficient.
To understand the factoring that follows, compare the following multiplications:
F O I L
(x + 2) (x + 5) = x2 + 5x + 2x + 2.5
= x2 + 7x + 10;
(x – 2) (x – 5) = x2 – 5x – 2x + 2.5
= x2 – 7x + 10;
(x + 3) (x – 7) = x2 – 7x + 3x + 3(-7)
= x2 – 4x – 21;
(x – 3) (x + 7) = x2 + 7x – 3x + (-3)7
= x2 + 4x -21.
Note that for all four products:
a. The product of the two binomials is a trinomial.
b. The coefficient of x in the trinomial is the sum of the constant terms in the binomials.
c. The constant term in the trinomial is the product of the constant terms in the binomials.
d. These observations lead to a method for factoring certain trinomials. The first
type we consider has a positive constant term, just as in the first two multiplications above.