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Difference of Squares

Reference > Mathematics > Algebra > Factoring Higher Degree Polynomials

In this unit we're going to explore different techniques for factoring polynomials with degree higher than two. If you are looking for information about factoring quadratics, you will find useful tutorials in this unit on factoring, this unit picks up where that unit left off, and will provide helpful techniques for factoring cubic polynomials, quartic polynomials, and more.

First, though, I want to review a quadratic factoring shortcut, because it'll come in handy in a later section of this unit. Let's suppose you have the quadratic x2 - 9, and you wanted to factor it. You could use the rules you learned in the previous unit, like this:

Rewrite the quadratic as x2 + 0x - 9. Now look for two numbers that add to 0 and multiply to -9. The two numbers are 3 and -3, therefore, this factors into (x - 3)(x + 3).

But that's not the quickest way to factor x2 - 9. Take a look at the following multiplication problem:

(a + b)(a - b)
a(a - b) + b(a - b)
a2 - ab + ab - b2
a2 - b2

We can turn this multiplication into a nice little factoring rule:

Difference of Squares
a2 - b2 = (a + b)(a - b)

This little factoring rule crops up over and over again in mathematics, so you should commit it to memory. Let's look at a few examples of how we can use it:

Example One
Factor 64x2 - 9

Both 64x2 and 9 are perfect squares, so this factors into (8x + 3)(8x - 3)

Example Two
Factor 27x2 - 12y2

First we note that 3 can be factored out, giving us 3(9x2 - 4y2). Since 9x2 and 4y2 are both perfect squares, this will then factor into 3(3x + 2y)(3x - 2y)

Example Three
Factor 16x4 - 81

Notice that now we are extending this to polynomials with degree higher than 2! Both 16x4 and 81 are perfect squares, so this factors into (4x2 + 9)(4x2 - 9)

You might think we're done at this point, but notice that the second factor is made up of two perfect squares with a minus between them; that's also a difference of squares! So the complete factorization is:

(4x2 + 9)(2x + 3)(2x - 3)


Factor x2 - 4
Factor 2x3 - 32x
Factor x4 - 9y2
Factor 9999 (think of it as 1002 - 12)
Factor 256x4 - 81y8
Factor 65,536x16 - 1
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