Factoring the Difference of Two Squares

Hello, and welcome to this video on factoring the difference of two squares.

When factoring polynomials, there are a few special patterns you’ll want to be on the lookout for. The one we’ll be talking about in this video is the difference of two squares.

When you see a binomial in the form \(a^{2}-b^{2}\), then you know you are looking at a difference of two squares, and it will factor to \((a+b)(a-b)\). The reason there’s no middle term is because FOILing these binomials will result in the outer and inner terms canceling each other out.

 
Let’s look at a few examples.

Example #1

 
We know this is an example of a difference of two squares because it is a binomial with a subtraction sign between the two terms, and both terms are perfect squares. To factor this binomial, take the positive square root of both terms (ignoring the minus sign).

 
Remember, a difference of squares binomial factors to \((a+b)(a-b)\). In this case, \(a=x\) and \(b=2\), so the factored form is:

 
If you wanted to check your answer, you can always FOIL it and see if you get the original expression.

Example #2

Let’s try another one.

 
First, check that this matches the pattern \(a^{2}-b^{2}\). It does, so take the square root of both terms.

 
Now, write out the factored form of the binomial.

Example #3

Let’s try one that’s slightly harder.

 
First, check that this matches the pattern \(a^{2}-b^{2}\). It looks different from our other examples, but this does follow this pattern because \(4y^{2}\) is a perfect square.

Take the positive square root of both terms.

 
Now we know that \(2y\) is the positive square root of \(4y^{2}\) because we’re going to take the square root of our coefficient 4, \(\sqrt{4}=2\), and then multiply it by \(y^{2}\), which is \(y\). So that’s how we simplify this.

 
Then, create the factored form of the binomial.

Example #4

Let’s work through one more example before we go.

 
First, make sure it follows the pattern \(a^{2}-b^{2}\). Since it does, take the positive square root of both terms. So remember, to take the square root of an expression like this \((16a^{4})\), take the square root of the coefficient first. \(\sqrt{16}=4\), and then \(\sqrt{a^{4}}=a^{2}\), so we’ll have \(4a^{2}\).

 
Then, write the factored form of the binomial.

 
And there you have it! Now you should be more comfortable recognizing difference of squares problems and how to factor them.

I hope this video was helpful. Thanks for watching, and happy studying!