1.3: Use remainders to determine and factorise a cubic

where $\displaystyle p$ and $\displaystyle q$ are constants. The remainder when $\displaystyle f(x)$ is divided by $\displaystyle (x-1)$ is $\displaystyle -12$. The remainder when $\displaystyle f(x)$ is divided by $\displaystyle (x-4)$ is $\displaystyle 30$.

(a) Find the value of $\displaystyle p$ and the value of $\displaystyle q$.

Using your values of $\displaystyle p$ and $\displaystyle q$,

(b) show that $\displaystyle f(3)=0$,

(c) express $\displaystyle f(x)$ as a product of linear factors,

(d) hence solve $\displaystyle f(x)=0$.