Divisibility of a rational function

Question

Problem

Determine coefficients $a$ and $b$ such that $$ \dfrac{x^3+ax^2+bx-6}{bx^2+2x+a} $$ is divisible by $x-2.$

• What is actually meant by divisible in this case?

Answer 1

What is actually meant by divisible in this case?

It's not very clear... You could call this fraction "divisible by $x-2$" if you can factor $x-2$ in the numerator, but I have the feeling they might mean that you can factor $x-2$ in the numerator and the denominator, and thus simplify the fraction.

In that case, you don't have to perform the division which is a bit annoying with the parameters $a$ and $b$. Recall that a polynomial is divisible by $x-c$ if $c$ is a root of the polynomial so with $p(x)=x^3+ax^2+bx-6$ and $q(x)=bx^2+2x+a$, you want $p(2)=0$ and $q(2)=0$. This will give you a linear system of two equations in the parameters $a$ and $b$.