Question

Find the values of p and q, if the polynomial $x^4+p{x}^3+2x^2-3x+q$ is divisible by the polynomial $x^2-1.$

Answer 1

$x^2-1$ is a factor
$\Rightarrow x^2-1=0$
(x+1)(x-1)=0
$\Rightarrow x+1=0\Rightarrow x=-1x-1=0\Rightarrow x=1$
-1 & 1 are zeroes
$p\left(x\right)=x^4+p{x}^3+2x^2-3x+qp(-1)=(-1{)}^4+p(-1{)}^3+2(-1{)}^2-3(-1)+q=0\Rightarrow 1-p+2+3+q=0q-p+6=0....(1)$
$p\left(1\right)=(1{)}^4+p(1{)}^3+2(1{)}^2-3(1)+q=0$
1+p+2-3+q=0
p+q=0 ...(2)
$\Rightarrow p=-q$
Substituting in (1)
q-p+6=0
q+q+6=0
2q=-6
q=-3
p=-q=-(-3)=3