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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Factoring is an essential skill for success in algebra and higher level mathematics courses. Therefore, we have taken great care in developing the student's understanding of the factorization process. The technique is consistently illustrated by displaying an empty set of parentheses and describing the thought process used to discover the terms that are to be placed inside the parentheses.The factoring scheme for special products is presented with both verbal and symbolic descriptions, since not all students can interpret symbolic descriptions alone. Two techniques, the standard "trial and error" method, and the "collect and discard" method (a method similar to the "ac" method), are presented for factoring trinomials with leading coefficients different from 1. Objectives of this module: be able to factor 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 b x 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 - 4 x - 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 - 4 x - 21 ( x - 7 ) ( x 3 )

The problem of factoring the polynomial a x 2 b x c , a≠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 + b x + c

Trial and error method

Consider the product

Steps showing the product of two binomials 'four x plus three,' and 'five x plus two.' See the longdesc for a full description.

Examining the trinomial 20 x 2 23 x 6 , we can immediately see some factors of the first and last terms.

20 x 2 6
20 x , x 6 , 1
10 x , 2 x 3 , 2
5 x , 4 x


Our goal is to choose the proper combination of factors of the first and last terms that yield the middle term 23 x .
Notice that the middle term comes from the sum of the outer and inner products in the multiplication of the two binomials.
The product of two binomials four x plus three, and five x plus two. The outer product of binomials is eight x, and the inner product is fifteen x.
This fact provides us a way to find the proper combination.

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

The proper combination we're looking for is

The product of the first and the last term is twenty x squared. One of the combinations of the factors of the first and last term yields two new factors of the product such that their sum is the middle term: twenty three x.

Sample set a

Factor 6 x 2 + x 12 .

Factor the first and last terms.

The factors of the first term 'six x squared' and the last term 'negative twelve' are shown. The product of the first and the last term is negative seventy-two x squared. One of the combinations of the factors of the first and the last term yields two new factors of the product such that their sum is the middle term: x.

Thus, 3 x and 3 are to be multiplied, 2 x and 4 are to be multiplied.

6 x 2 + x 12 = ( ) ( ) Put the factors of the leading term in immediately . = ( 3 x ) ( 2 x ) Since 3 x and 3 are to be multiplied, they must be located in different binomials . = ( 3 x ) ( 2 x + 3 ) Place the 4 in the remaining set of parentheses . = ( 3 x 4 ) ( 2 x + 3 ) 6 x 2 + x 12 = ( 3 x 4 ) ( 2 x + 3 )

C h e c k : ( 3 x 4 ) ( 2 x + 3 ) = 6 x 2 + 9 x 8 x 12 = 6 x 2 + x 12

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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