**Factoring x ^{2} + bx + c**

We now begin a study of the factoring of trinomials. We first factor trinomials like

x^{2}+ 5x + 6 and x^{2} + 3x – 10

by a refined *trial-and-error process*. In this section, we restrict our attention to trinomials of the type ax^{2} + bx + c where a = 1. The coefficient *a *is often called the **leading coefficient.**

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To understand the factoring that follows, compare the following multiplications:

F O I L

(x + 2) (x + 5) = x^{2} + 5x + 2x + 2.5

= x^{2} + 7x + 10;

(x – 2) (x – 5) = x^{2} – 5x – 2x + 2.5

= x^{2} – 7x + 10;

(x + 3) (x – 7) = x^{2} – 7x + 3x + 3(-7)

= x^{2} – 4x – 21;

(x – 3) (x + 7) = x^{2} + 7x – 3x + (-3)7

= x^{2} + 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.