Question

If $5x^3+5x^2-6x+9$ is divided by $(x+3)$, find the remainder.

Answer 1

Determine the remainder.

If $p\left(x\right)$ is divided by $x+c$, then the remainder is $p\left(-c\right)$.$[\because x+c=x-(-c\left)\right]$

The divisor here is $x+3$.

$\begin{array}{rcl}x+3& =& x-\left(-3\right)\left[\mathrm{Form}x-c\mathrm{with}c=-3\right]\end{array}$

Let $f\left(x\right)=5x^3+5x^2-6x+9$

By remainder theorem, when $f\left(x\right)$ is divided by $x+3$, then the remainder is $f\left(-3\right)$.

$\begin{array}{rcl}f\left(-3\right)& =& 5{\left(-3\right)}^3+5{\left(-3\right)}^2-6\left(-3\right)+9\\ & =& -135+45+18+9\\ & =& -63\end{array}$.

Hence, the remainder is $-63$.