Question

In each of the following, using the remainder theorem, find the remainder when $f\left(x\right)$ is divided by $g\left(x\right)$ and verify the result by actual division:

1. $f\left(x\right)=x^3+4x^2-3x+10,g\left(x\right)=x+4$

Answer 1

Given: $f\left(x\right)=x^3+4x^2-3x+10$ is divided by $g\left(x\right)=x+4$.

By remainder theorem, if polynomial $f\left(x\right)$ is divided by the linear polynomial $(x-a)$; then remainder is equal to $f\left(a\right)$.

$\therefore f(-4)=(-4{)}^3+4\times (-4{)}^2-3\times (-4)+10$

$=-64+64+12+10$

$=22$

Now, verify the result by long division method.

$\therefore (x^3+4x^2-3x+10)÷(x+4)$

$x+4)x^3+4x^2-3x+10(x^2-3x$
$-$
$x^3+4x^2$


$-3x+10$
$-$
$-3x-12$

$22$

Hence, the reamainder is $22$ verified.