In each of the following, using the remainder Theorem, find the remainder when $f(x)$ is divided by $g(x)$ and verify the result by actual division.$f(x) = x^3 + 4x^2 - 3x + 10, g(x) = x + 4$

Given:

$f(x) = x^3 + 4x^2 - 3x + 10, g(x) = x + 4$

To do:

Use remainder theorem to find the remainder when f(x) is divided by g(x).

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 $p(a)$.

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

$g(x) = x + 4$

$=x - (-4)$

So, the remainder will be $f(-4)$.

$f(-4) = (-4)^3 + 4(-4)^2 - 3(-4) + 10$

$= -64 + 4(16) + 12 + 10$

$= -64+64+22$

$=22$

Therefore, the remainder is $22$.