Expressions, Relations, and Functions - Beginning Pre-Calculus for Game Developers - page 56

Basic Factoring
In an earlier chapter, you explored the distributive property of numbers. As a
refresher, here is an example of the applications of the principle of distribution to
a multiplication problem:
4 ð 5 þ 3 Þ¼ 4 ð 5 Þþ 4 ð 3 Þ¼ 20 þ 12 ¼ 32
4 ð 5 3 Þ¼ 4 ð 5 Þ 4 ð 3 Þ¼ 20 12 ¼ 8
To perform the distributions, you begin by evaluating the terms in parentheses in
relation to the numbers that are applied to them. You can then rearrange the
terms so that you preserve the operators that characterize the relations between
them. You can rearrange the terms because 4 constitutes a number that is
common to each term. You distribute the multiplication activities of 4 so that
you apply them separately to the numbers within the parentheses.
When you factor the terms of an expression, you reverse this activity. You begin
with a situation in which a common number is applied to a set of terms. You then
rewrite the expression so that you combine the common terms into groups. Here
is how you factor the expressions shown previously:
4 ð 5 Þþ 4 ð 3 Þ¼ 4 ð 5 þ 3 Þ
4 ð 5 Þ 4 ð 3 Þ¼ 4 ð 5 3 Þ
These expressions both have 4 distributed across multiplication operations
involving 3 and 5. To factor the expressions 4(5) and 4(3), you observe that 4
constitutes a common term. You can then factor this common term so that you
apply it to the other terms in a collective way.
When you factor out a term, you usually try to factor out the largest common
factor. The largest common factor represents other factors combined. Consider
this equation:
14 a 56
You can rewrite this expression so that each expression consists of the lowest
common factors:
ð 1 Þð 2 Þð 7 Þa ð 1 Þð 2 Þð 2 Þð 2 Þð 7 Þ