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(5 + 3) = 4(5) + 4(3) = 20 + 12 = 32
4(5 – 3) = 4(5) – 4(3) = 20 – 12 = 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(5) + 4(3) = 4(5 + 3)
4(5)– 4(3) = 4(5 – 3)
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:
14a – 56
You can rewrite this expression so that each expression consists of the lowest common factors:
(1)(2)(7)a – (1)(2)(2)(2)(7)
You are then in a position to group the common factors so that they reveal the largest common factor:
[(1)(2)(7)]a – [(1)(2)](2)(2)[(7)]
If you carry out the implied multiplication, you arrive at the largest common factor for the two terms:
(14)a –(14)(2)(2) = 14a – 14(4) = 14(a – 4)
One of the key notions in factoring is that when you factor an expression, you end up with a product. As Figure 4.1 illustrates, factoring two expressions results in a new expression that implies that a multiplication can take place. Generally, then, you have successfully factored an expression when you rewrite it as a product.
Figure 4.1. Factoring results in a product.
Given that you factor a term, you can then check the correctness of your activities if you carry out the implied multiplication:
14(a – 4) = 14a – 56
A further extension of factoring involves collecting like terms. If an expression contains terms that are exactly alike, then you can rewrite the expression so that you use one instead of several instances of the like term in the expression. As in previous examples, when you collect like terms, you create a product. As examples of expressions possessing collectable terms, consider the following:
8c + 6c = (8 + 6)c = 14c
7a2 + 5a2 + 6a3 – 3a3 = (7 + 5)a2 +(6 – 3)a3 = 12a2 + 3a3 = 3(4a2 + a3)
In the first example, c is common to both terms, so you can use the distributive property to factor the sum of 8 and 6 into a single expression that you can multiply by c. After you regroup the integers, you can carry out the addition and arrive at a single term, 14c.
In the second expression, you perform slightly more involved activities. The expression contains two like terms, a2 and a3. You can group the coefficients of these terms into expressions that involve addition and subtraction. When you carry out these operations, you end up with 12 and 3 as coefficients. You then carry the process a step further by factoring 3 out of the expression.
Exercise Set 4.1Here are a few problems for factoring. Find the largest common factor or the like terms.
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