These trinomials differs from those in the previous section (x^{2}+bx+c) in that the coefficient of x^{2} is not 1.

To factor a trinomial of the form ax^{2}+bx+c means to express the polynomial as the product of two binomials. There are various methods of factoring these trinomials. The method described below is factoring trinomials using trial factors. Factoring such polynomials by trial and error may require testing many trial factors.

To Factor a trinomial using trial and error:

Find two terms whose product is ax^{2}. These terms are the first terms of the trial binomial factors.

Find two terms whose product is c. These terms are the last terms of the trial binomial factors.

Use FOIL to find the algebraic sum of the products of the outer and inner terms in the trial binomial factor. If this algebraic sum does not equal the middle term, bx, try all pairs of terms that give the products ax^{2} and c of the trinomial. If all fail, then the trinomial is not factorable.

To reduce the number of trial factors, remember the following points:

Points to remember in Factoring ax^{2}+bx+c

If the terms of the trinomial do not have a common factor, then the terms of a binomial factor cannot have a common factor.

When the constant term of the trinomial is positive, the constant terms of the binomials have the same sign as the coefficient of x in the trinomial.

When the constant term of the trinomial is negative, the constant terms of the binomials have opposite signs.

Factor 3x^{2}-2x-8

Find the factors of a (3) and the factors of c (-8).
Because the constant term, c, of the trinomial is negative (-8), the constant terms of the binomials factors will have opposite sign

Factors of 3

Factors of -8

1,3

2,-4

-2,4

Using these factors, write trial factors, and use the Outer and Inner products of FOIL to check the middle term.