Factoring the Difference of Two Squares

Factoring the Difference of Two Squares Video

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.

\(a^{2}-ab+ab-b^{2}\)
 
\(a^{2}-b^{2}\)

 

Let’s look at a few examples.

\(x^{2}-4\)

 

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).

\(\sqrt{x^{2}}=x\)
 
\(\sqrt{4}=2\)

 

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:

\((x+2)(x-2)\)

 

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

Let’s try another one.

\(x^{2}-36\)

 

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

\(a=\sqrt{x^{2}}=x\)
 
\(b=\sqrt{36}=6\)

 

Now, write out the factored form of the binomial.

\((x+6)(x-6)\)

 

Let’s try one that’s slightly harder.

\(4y^{2}-16\)

 

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.

\(a=\sqrt{4y^{2}}=2y\)

 

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.

\(b=\sqrt{16}=4\)

 

Then, create the factored form of the binomial.

\((2y+4)(2y-4)\)

 

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

\(16a^{4}-49\)

 

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 \(4a^{2}\).

\(a=\sqrt{16a^{4}}=4a^{2}\)
 
\(b=\sqrt{49}=7\)

 

Then, write the factored form of the binomial.

\((4a^{2}+7)(4a^{2}-7)\)

 

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!

 

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by Mometrix Test Preparation | This Page Last Updated: October 13, 2023