# Difference of Squares

Reference > Mathematics > Algebra > Factoring Higher Degree PolynomialsIn 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 x^{2} - 9, and you wanted to factor it. You could use the rules you learned in the previous unit, like this:

*Rewrite the quadratic as x ^{2} + 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 x^{2} - 9. Take a look at the following multiplication problem:

(a + b)(a - b)

a(a - b) + b(a - b)

a^{2} - ab + ab - b^{2}

a^{2} - b^{2}

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

**Difference of Squares**

a^{2} - b^{2} = (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 64x^{2} - 9

**Solution**

Both 64x^{2} and 9 are perfect squares, so this factors into (8x + 3)(8x - 3)

**Example Two**

Factor 27x^{2} - 12y^{2}

**Solution**

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

**Example Three**

Factor 16x^{4} - 81

**Solution**

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

(4x^{2} + 9)(2x + 3)(2x - 3)

## Questions

^{2}- 4

^{3}- 32x

^{4}- 9y

^{2}

^{2}- 1

^{2})

^{4}- 81y

^{8}

^{16}- 1