Question

Use remainder theorem to find remainder when $p\left(x\right)$ is divided by $q\left(x\right)$ in the following questions:$p\left(x\right)=x^4+x^3+x^2-5x+1,q\left(x\right)=x+1$

Answer 1

Given, $p\left(x\right)=x^4+x^3+x^2-5x+1$ and $g\left(x\right)=x+1$
$q\left(x\right)\Rightarrow$ $x+1=0$ $\Rightarrow$$x=-1$

The remainder when $p\left(x\right)$ is divided by $g\left(x\right)$ is $p(-1)$ ...(By Remainder Theorem)
Thus,
$p(-1)=(-1{)}^4+(-1{)}^3+(-1{)}^2-5(-1)+1$
$p(-1)=1-1+1+5+1$
$p(-1)=7$