Find the remainder when $x^3+ 3x^2 + 3x + 1$ is divided by $ x+\pi $

Given:

$x^3+ 3x^2 + 3x + 1$ is divided by $x+\pi$

To do:

We have to find the remainder when $x^3+ 3x^2 + 3x + 1$ is divided by $x+\pi$.

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 $f(x) = x^3+ 3x^2 + 3x + 1$ and $g(x) = x+\pi=x-(-\pi)$

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

$f(-\pi) = (-\pi)^3+3(-\pi)^2+3(-\pi) + 1$

$= -pi^{3}+3\pi^{2}-3\pi+1$

Therefore, the remainder is $-pi^{3}+3\pi^{2}-3\pi+1$.