Question

Find the remainder when $x^3-a{x}^2+6x-a$ is divisible by $\left(x-a\right)$.

Answer 1

Find the remainder by using the remainder theorem:

Here, divident is $x^3-a{x}^2+6x-9$ and the divisor is $\left(x-a\right)$.

Let the polynomial $p\left(x\right)=x^3-a{x}^2+6x-a...\left(1\right)$

Put Divisor$=0$.

$$\begin{gathered} \Rightarrow x-a=0 \\ \Rightarrow x=a \end{gathered}$$

Theorem:

Remainder theorem states that when $f\left(x\right)$ is divided by $\left(x-a\right)$, then the remainder is $f\left(a\right)$ .

$\begin{array}{rcl}& \Rightarrow & p\left(a\right)=\left(a^3\right)-a{\left(a\right)}^2+6a-a\\ & =& a^3-a^3+6a-a\\ & =& 6a-a\\ & =& 5a\end{array}$

Hence, the remainder is $5a$.