Find the remainder when $ x^{3}-a x^{2}+6 x-a $ is divided by $ x-a $.

To do:

We have to find the remainder when \( x^{3}-a x^{2}+6 x-a \) is divided by \( x-a \).

Solution:

The remainder theorem states that when a polynomial $p(x)$ is divided by a linear polynomial, $x - a$ the remainder of that division will be equivalent to $p(a)$.

Let $f(x) =x^{3}-a x^{2}+6 x-a$ and $p(x) = x -a$

This implies,

The remainder will be $f(a)$.

$f(a) =(a)^{3}-a (a)^{2}+6 (a)-a$

$= a^3-a^3+6a-a$

$=5a$

Therefore, the remainder is $5a$.