Lane ORCCA (2020–2021): Open Resources for Community College Algebra

Subsection 7.2.1 Factoring out Common Polynomials

Recall that to factor \(3x+6\text{,}\) we factor out the common factor \(3\text{:}\)

The “\(3\)” here could have been something more abstract, and it still would be valid to factor it out:

In fact, even “larger” things can be factored out, as in this example:

In this last example, we factored out the binomial factor \((a+b)\text{.}\) Factoring out binomials is the essence of this section, so let's see that a few more times:

Even with an expression like \(Q^2(Q-3)+Q-3\text{,}\) if we re-write it in the right way using a \(1\) and some parentheses, then it too can be factored:

The truth is you are unlikely to come upon an expression like \(x(x+2)+3(x+2)\text{,}\) as in these examples. Why wouldn't someone have multiplied that out already? Or factored it all the way? So far in this section, we have only been looking at a stepping stone to a real factoring technique called factoring by grouping.

Subsection 7.2.2 Factoring by Grouping

Factoring by grouping is a factoring technique that sometimes works on polynomials with four terms. Here is an example.

Unfortunately, this technique is not guaranteed to work on every polynomial with four terms. In fact, most randomly selected four-term polynomials will not factor using this method and those selected here should be considered “nice.” Here is an example that will not factor with grouping:

In this example, at the step where we hope to see the same binomial appearing twice, we see two different binomials. It doesn't mean that this kind of polynomial can't be factored, but it does mean that “factoring by grouping” is not going to help. This polynomial actually factors as \((x+1)(x+2)(x+3)\text{.}\) So the fact that grouping fails to factor the polynomial doesn't tell us whether or not it is prime.