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

The correct option is A: $0$

Given polynomial is $f(x)=x^{3}+3x^{2}+3x+1$

It has to be divided by $x+1$.

Then, $x+1=0$

or, $x=-1$

By Remainder theorem, $f(-1)$ will be the remainder.

Put the value $x=-1$ in $f(x)$, we get

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

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

$ f(-1)=0$

So, the remainder is $0$, when polynomial $x^{3}+3x^{2}+3x+1$ is divided by $x+1$.