## Patterns When Factoring

When factoring, there are some specific patterns that come up from time to time that can be useful to know. Here we’re going to take a look at a few of these patterns and their applications. Let’s start with factoring the difference of squares.

$a^2 - b^2 = (a+b)(a-b)$

This is a tricky one if you haven’t ever considered it and run into a problem asking you to factor something of the form a² – b². This is kind of pretty, and makes certain things really easy to use. With polynomials as pertains to Algebra 1, you might have to factor something like this at some point:

$24x^2y^2 - 54$

The first step is to see if any factoring can be done using the GCF. The GCF of 24x²y² and 54 is 6, so that gives us the following:

$6(4x^2y^2 - 9)$

At this point it can be tricky to know what to do next, until you realize that both 4x²y² and 9 are perfect squares. Then we get this using the formula for our pattern above:

$6(2xy + 3)(2xy - 3)$

And we have the original polynomial factored. As always, you should check your answer by multiplying these factors together. We’ll leave that as an exercise.

Another pattern that comes up is the square of a binomial. There are two possibilities depending on whether the operation in the binomial is addition or subtraction. Here are the two patterns:

$(a+b)^2 = a^2 + 2ab + b^2$
$(a-b)^2 = a^2 - 2ab + b^2$

If you’re so inclined, you can multiply out these binomials by hand to make sure that the patterns do what they say they do.

The first of these two patterns offers a glimpse into a mental math shortcut for finding the square of a two-digit number. Consider the process of finding the square of 23:

$23 = (20+3)^2 = 20^2 + 2(20)(3) + (3)^2$
$400 + 120 + 9$
$529$

This takes a little bit of practice, but is fun to be able to do.