6.7 Factoring trinomials with leading coefficient other than 1

Overview

• The Method of Factorization

The method of factorization

In the last section we saw that we could easily factor trinomials of the form $x^2+bx+c$ by finding the factors of the constant $c$ that add to the coefficient of the linear term $b$ , as shown in the following example:

Factor $x^2-4x-21$ .
The third term of the trinomial is $-21$ . We seek two numbers whose

(a) product is $-21$ and
(b) sum is $-4$ .

The required numbers are $-7$ and $+3$ .

$x^2-4x-21=(x-7)(x+3)$

The problem of factoring the polynomial $a{x}^2+bx+c$ , $\mathrm{a\ne 1}$ , becomes more involved. We will study two methods of factoring such polynomials. Each method produces the same result, and you should select the method you are most comfortable with. The first method is called the trial and error method and requires some educated guesses. We will examine two examples (Sample Sets A and B). Then, we will study a second method of factoring. The second method is called the collect and discard method , and it requires less guessing than the trial and error method. Sample Set C illustrates the use of the collect and discard method.

The trial and error method of factoring $a{x}^2+bx+c$

Trial and error method

Look for the combination that when multiplied and then added yields the middle term.

The proper combination we're looking for is

Sample set a