Factor theorem and factorisation of a cubic

Statement

Let $f(x)=x^3-4x^2+x+6$.

• (a) Verify, using the factor theorem, that $(x-2)$ is a factor.
• (b) Fully factorise $f(x)$ and solve $f(x)=0$.

Solution

(a) $f(2)=8-16+2+6=0$, so by the factor theorem $(x-2)$ divides $f(x)$.

(b) Dividing: $f(x)=(x-2)(x^2-2x-3)=(x-2)(x-3)(x+1)$. The solutions of $f(x)=0$ are $x=-1,\ 2,\ 3$.

$\boxed{f(x)=(x+1)(x-2)(x-3);\quad x=-1,\ 2,\ 3}$