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How to find the difference of squares

One of the common questions I get asked when I am tutoring algebra is how to find the difference of squares. First, what exactly is a difference of squares, and what is it used for? Second, how do you find it?

The difference of squares is a tool used to factor certain types of polynomials. Factoring is often useful in simplifying equations and allow some form of cancellation or combination of like factors. The difference of squares will allow you to factor some polynomial types that are not otherwise factorable, making it a useful tool in algebra and anything that uses algebra.

So how do you find it? You can find the difference of squares for any polynomials which is a difference of two perfect squares. Take the simplest case: x2 -1. This polynomial is the difference of two perfect squares: x2 is, obviously, the square of x while 1 is the square of 1. The resulting factors, using the difference of squares, is (x+1)(x-1).
 
To confirm that this is, in fact, the factors of the polynomial, let's multiply it out using the FOIL method. FOIL will yield the equation x2-x+x-1. The positive x and the negative x cancel each other out, leaving us with x2 - 1, which is our original polynomial. The cancelling of the values with a single x are the key to why the difference of squares works.
 
Now, let's take a step back from this simple case and see if we can find a general solution. Given a polynomial a- b2, we can identify it as a difference of squares. The two squares are a and b, and the polynomial is the difference between their squares. Looking at the pattern from our earlier example, we can see that the factor will be (a+b) and (a-b). This will work for the difference of ANY two perfect squares.
 
What if it doesn't originally look like a square? If we have the polynomial x4-y6, for example, that would appear to be a difference of a 4th power and a sixth power. However, if we look at is a little sideways, we can see that this actually is a difference of squares. x4 is actually the square of x2, while y6 is actually the square of y3. This means that our value for a is x2 and b is y3. Plugging those into our factors, we find that the factors of the equation x4-y6 are (x2 + y3) and (x2 - y3).
 
If we check it with FOIL, we get x4 + x2y3 - x2y3 - y6. The inner factors cancel out, leaving us with our original polynomial x- y6.
 
This works the other way as well. What if our original polynomial is x - 1? Since x is actually also a perfect square (it is the square of the square root of x) we can still factor this using the difference of squares method. The result is (√x +1) and (√x -1). Feel free to check it.

The difference of squares is a useful tool when we are trying to factor equations. It might not always seem useful; why would we want to factor x-1? However, if somewhere down the line we can cancel it with a factor from another polynomial (for example, (√x +1)), we can use those factors to simplify the equation and make our life easier down the road.