Question

Find the remainder when x³ + 3x² + 3x + 1 is divided by  x + 1.

Answer 1

Let us denote the given polynomials as

${\displaystyle f\left(x\right)=x^3+3x^2+3x+1,}$

${\displaystyle g\left(x\right)=x+1}$

${\displaystyle \Rightarrow g\left(x\right)=x-(-1),}$

We will find the remainder when f(x) is divided by g(x).

By the remainder theorem, when f(x) is divided by g(x) the remainder is

${\displaystyle =f(-1)}$

${\displaystyle ={(-1)}^3+3(-1)+1}$

${\displaystyle =-1+3-3+1}$

${\displaystyle =0}$