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

Given: $x^3+x^2-x+1$ is divided by $x+2$.

To do: To find the remainder.

Solution:

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

Let $x+2=0$

$\Rightarrow x=-2$, put this value in $f( x)=x^3+x^2-x+1$

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

$=-8+4+2+1$

$=-1$

Thus, the remainder is $-1$, when $x^3+x^2-x+1$ is divided by $x+2$.