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Monomial Factors

The first thing you should know before factoring an algebraic expression is to identify any factors which are monomials.

Let's see what this is all about in the following example.

Example:

Remove all common monomial factors from 42b 2y – 28by 2.

solution:

This example is very similar to the previous one, and so you should try it as a practice problem on your own first, before looking at our brief outline of a solution. First, write out the factorization of each of the two terms explicitly:

42b 2y = 2 · 3 · 7 · b 2 · y 1

and

28by 2 = 2 2 · 7 · b 1 · y 2

Notice that we’ve written exponents of symbols explicitly, even if those exponents are 1 (just as a visual cue when we check now for common factors between the two terms). It is also helpful to sort the factors of the individual terms in a common order. Here numerical prime factors are sorted from smallest to largest (going left to right) and symbolic factors are sorted alphabetically.

Comparison of these two factorizations indicates immediately that the common factors are 2, 7, b, and y, all to the first power. Thus

42b 2y – 28by 2 = 2 · 7 · b · y (3b – 2y)

= 14by (3b – 2y).

The terms in the expression in brackets on the right here are obtained by taking what’s left of each of the original terms when the common factors are removed. Thus, since

42b 2y = 2 · 3 · 7 · b 2 · y 1

when we remove the factors 2, 7, b 1 , and y 1, all that’s left is the 3 and one of the factors b, or 3b. A similar inspection indicates that after removal of these four common factors from 28by 2, all that is left is the factors 2 and y, each to the first power.

Thus, the required factorization here is

42b 2y – 28by 2 = 14by (3b – 2y).

We’ll leave it up to you to verify that this is correct by multiplication.

 
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