Question

How do you simplify `\frac { q - 16} { q ^ { 2} - 18q + 32}`?

Answer 1

The simplified answer is `1/(q-2)`.

Explanation:

First, factor the polynomial `q^2-18q+32`.

`-16` and `-2` are a factor pair of `32` that also add up to `-16`
(i.e. `-16*-2=32` and `-16+(-2)=-18`). So these will be our zeroes in our factored equation:

`q^2-18q+32=>(q-16)(q-2)`

Now, let's put this back into the original problem:

`(q-16)/(q^2-18q+32)=>color(red)(q-16)/(color(red)((q-16))(q+2))`

The two factors highlighted in red can now be cancelled out, revealing the answer:

`cancel(q-16)/(cancel((q-16))(q-2))=>1/(q-2)`

We can check our answer is correct by checking if the graphs are the same. Here's the graph of `(q-16)/(q^2-18q+32)`: graph{(x-16)/(x^2-18x+32) [-10, 15, -6, 6]}
And here's the graph of `1/(q-2)`: graph{1/(x-2) [-10, 15, -6, 6]}
Since the graphs are identical, the polynomials are equal, so our answer is correct.