Find the possible values of a in the cubic equation.

Question

Given that $(x-a)$ is a factor of $x^3-ax^2+2x^2-5x-3$, find the possible values of the constant $a$.

I believe you first have to find the $a$ in the cubic equation then the other $a$ in $(x-a)$, but I'm having problems finding the first $a$. So far all I have is:

$x^3-ax^2+2x^2-5x-3=0$

$x^3-2ax^2-5x=3$

Answer 1

If $(x - a)$ is a factor of a polynomial $p(x)$, then $p(a) = 0$. In this problem, we must therefore have the equation $$a^3 - a(a^2) + 2a^2 - 5a - 3 = 0.$$ You should be able to find the possible values of $a$ from that.

Answer 2

Hint: If $x-a$ is a factor, then $a$ is a zero. Hence, $$a^3-a^3+2a^2-5a-3=0\implies 2a^2-5a-3=0.$$