Question

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

Answer 1

Let $f\left(x\right)=x^3+3x^2+3x+1$.

The polynomial $f\left(x\right)$ is divided by $x+1$.

Then $x+1=0⟹x=-1.$
By remainder theorem, the remainder is given by $f(-1)$.
Hence, remainder of the given polynomial $f\left(x\right)$ is
$f(-1)=(-1{)}^3+3(-1{)}^2+3(-1)+1$ $=-1+3-3+1=0$.
That is, the remainder obtained is $0$.