if a is in Q how would i show that f(a) is the remainder after dividing the polynomial f(x) by (x-a)?
If $\displaystyle f(x)$ is a polynomial of degree $\displaystyle n $ in $\displaystyle x$, dividing by $\displaystyle (x-a)$ leaves a quotient $\displaystyle G(x)$ of degree $\displaystyle n-1$ with remainder $\displaystyle R$ (a constant). That is $\displaystyle f(x) = (x-a)g(x)+R$. If you put $\displaystyle x = a$, you will get $\displaystyle f(a) = R$.