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.
