Page 138 - MarceAlgebra Demystified

CHAPTER 6 Factoring 125

The associative and distributive properties can be confusing. The associative
property states ðabÞc ¼ aðbcÞ. This property says that when multiplying three
(or more) quantities you can multiply the first two then the third or multiply
the second two then the first. For example, it might be tempting to write
5ðx þ 1Þðy 3Þ¼ð5x þ 5Þð5y 15Þ. But ð5x þ 5Þð5y 15Þ¼½5ðx þ 1ފ
½5ðy 3ފ ¼ 25ðx þ 1Þðy 3Þ. The ‘‘5’’ can be grouped either with ‘‘x þ 1’’
or with ‘‘y 3’’ but not both: ½5ðx þ 1ފðy 3Þ¼ð5x þ 5Þðy 3Þ or
ðx þ 1Þ½5ðy 3ފ ¼ ðx þ 1Þð5y 15Þ.
Factors themselves can have more than one term. For instance 3ðx þ 4Þ
xðx þ 4Þ has x þ 4 as a factor in each term, so x þ 4 can be factored from
3ðx þ 4Þ and xðx þ 4Þ:

3ðx þ 4Þ xðx þ 4Þ¼ð3 xÞðx þ 4Þ:

Examples

2xð3x þ yÞþ 5yð3x þ yÞ¼ð2x þ 5yÞð3x þ yÞ

10yðx yÞþ x y ¼ 10yðx yÞþ 1ðx yÞ¼ð10y þ 1Þðx yÞ

8ð2x 1Þþ 2xð2x 1Þ 3yð2x 1Þ¼ð8 þ 2x 3yÞð2x 1Þ

Practice

1: 2ðx yÞþ 3yðx yÞ¼

2: 4ð2 þ 7xÞ xð2 þ 7xÞ¼

3: 3ð3 þ xÞþ xð3 þ xÞ¼

4: 6xð4 3xÞ 2yð4 3xÞ 5ð4 3xÞ¼

5: 2x þ 1 þ 9xð2x þ 1Þ¼
4 4
6: 3ðx 2yÞ þ 2xðx 2yÞ ¼

Solutions

1: 2ðx yÞþ 3yðx yÞ¼ð2 þ 3yÞðx yÞ