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

Given:

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

To do:

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

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 $g(x) = x + 1 = x-(-1)$

So, the remainder will be $f(-1)$.

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

$= -1 + 3(1) -3+1$

$=3-3$

$=0$

Therefore, the remainder is $0$.