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!
