Question

Divide.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

Answer 1

It's not clear whether the problem is to:

(1) divide $x^2-4$ by $\dfrac{x-8}{x-2}$, or

(2) divide $\dfrac{x^2-4}{x-8}$ by $x-2$.

But with only two possible cases, that's not much work to do.

$\drac{x^2-4}{\frac{x-8}{x-2}}=\dfrac{(x^2-4)(x-2)}{x-8}=\dfrac{x^3-2x^2-4x+8}{x-8}$

Synthetic division yields a quotient and remainder of $x^2+6x+44+\dfrac{360}{x-8}$ (see attachment for how the algorithm was carried out).

In the other case,

$\dfrac{\frac{x^2-4}{x-2}}{x-8}=\dfrac{\frac{(x-2)(x+2)}{x-2}}{x-8}=\dfrac{x+2}{x-8}$

provided that $x\neq2$. Synthetic division can be used again here, but this is quite easy to deal with by just doing some easy manipulation of the numerator:

$\dfrac{x+2}{x-8}=\dfrac{x-8+10}{x-8}=\dfrac{x-8}{x-8}+\dfrac{10}{x-8}=1+\dfrac{10}{x-8}$

(also provided that $x\neq 8$)