# 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.

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

$$a^{2}-b^{2}$$

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