Question

How do you simplify `(x^2-4)/(x-3) * (x+2)/(8x-16)`?

Answer 1

`(x^2+4x + 4)/(8x-24)`

Explanation:

We should start by factoring the numerators and denominators separately and then seeing what we might be able to cancel out. The four terms we have are;

`x^2-4`
`x-3`
`x+2`
`8x-16`

`x-3` and `x+2` can't really be factored any further, but the other two can. Let's start with `x^2-4`. For a binomial expression of the form `x^2 - a^2`;

`x^2-a^2 = (x-a)(x+a)`

In our case, `a=2`, so;

`x^2-4 = x^2-2^2=(x-2)(x+2)`

Now take a look at `8x-16`. We can factor out the `8`, leaving;

`8(x-2)`

We can now rewrite the original expression as;

`((x-2)(x+2)(x+2))/((x-3)8(x-2))`

The only term that cancels out is `x-2`.

`(cancel(x-2)(x+2)(x+2))/((x-3)8cancel(x-2))`

When we multiply the remaining factors together, we get;

`(x^2+4x + 4)/(8x-24)`