Question Video: Simplifying Quotients of Algebraic Fractions by Factorization Mathematics

Video Transcript

Simplify the function 𝑛 of 𝑥 equals 𝑥 squared minus 16 over two 𝑥 squared plus nine 𝑥 divided by nine 𝑥 squared minus 72𝑥 plus 144 divided by four 𝑥 squared minus 81.

We’ll start with the first fraction, 𝑥 squared minus 16 over two 𝑥 squared plus nine 𝑥. We know that we don’t divide fractions. Instead, we multiply by the reciprocal. We flipped the numerator and the denominator. The next thing we want to do is see if we can simplify these expressions before we multiply them together. First, we’ll want to see if we can simplify 𝑥 squared minus 16. 𝑥 squared minus 16 reminds me of the difference of squares. I could write 16 as four squared. And the difference of squares can be written as 𝑎 plus 𝑏 times 𝑎 minus 𝑏, which means we can rewrite our numerator as the factors 𝑥 plus four times 𝑥 minus four.

Next, we wanna try and factor the denominator, two 𝑥 squared plus nine 𝑥. These two terms have a common factor of 𝑥. And so we can take out the 𝑥 term. If we divide two 𝑥 squared by 𝑥, we’re left with two 𝑥. And nine 𝑥 divided by 𝑥 equals nine. We can rewrite the first denominator as 𝑥 times two 𝑥 plus nine. Four 𝑥 squared minus 81, this again should remind us of the difference of squares. We could write four 𝑥 squared as two 𝑥 squared. And we could write 81 as nine squared. And that means we can rewrite four 𝑥 squared minus 81 as two 𝑥 plus nine times two 𝑥 minus nine.

Last term, when we look at nine 𝑥 squared minus 72𝑥 plus 144, we notice that all of the coefficients are divisible by nine. And that means we can take out the nine factor. And we would be left with 𝑥 squared minus eight 𝑥 plus 16. And looking at 𝑥 squared minus eight 𝑥 plus 16, it fits the form 𝑥 squared minus two 𝑏𝑥 plus 𝑏 squared. And that can be simplified to 𝑥 minus 𝑏 squared. We could rewrite negative eight as negative two times four. And we could rewrite 16 as four squared. We could then reduce that to 𝑥 minus four squared, nine times 𝑥 minus four squared. However, in this case, because we’re trying to simplify, it might be better for us to write it as nine times 𝑥 minus four times 𝑥 minus four. From here, we’ll take all the factorized forms and we’ll plug them back in, and our last factorized form.

Once we get to this point, we’re ready to see if anything cancels out. Are there any terms that are in the numerator and the denominator? We have a negative four in the numerator and the denominator; they cancel out. We have a two 𝑥 plus nine in the numerator and a two 𝑥 plus nine in the denominator, and they cancel out. What’s remaining in our numerator is 𝑥 plus four times two 𝑥 minus nine. And what’s remaining in our denominator, 𝑥 times nine times 𝑥 minus four.

It might seem really tempting to try and cross out 𝑥 plus four and 𝑥 minus four. But these are different values. They’re not the same in the numerator and the denominator. So they have to stay. 𝑛 of 𝑥 is the function we were looking for. And it’s equal to 𝑥 plus four times two 𝑥 minus nine over — to make the denominator more simplified, we can multiply nine and 𝑥 and write it as nine 𝑥 times 𝑥 minus four. This is our simplified function.