Question

Find the zeroes of the polynomials.
$x^3+2x^2-x-2$

Answer 1

Let $p\left(x\right)=x^3+2x^2-x-2$
The coefficient of the leading term is $1$ and the count is $-2.$
Also the factors of $2$ are $\pm 1$ and $\pm 2.$
So the possible integral zeroes of $p\left(x\right)$ are $\pm 1,\pm 2.$
Let us try for $x=1,$
$p\left(1\right)=(1{)}^3+2(1{)}^2-\left(1\right)-2$
$=1+2-1-2=0$
$\therefore 1$ is an integral zero of $p\left(x\right).$
Also,
$p(-1)=(-1{)}^2+2(-1{)}^2-(-1)-2$
$=-1+2+1-2=0$
Similarly for $x=-2$
$p(-2)=(-2{)}^3+2(-2{)}^2-(-2)-2$
$=-8+8+2-2=0$
$\Rightarrow -2$ is also an integral zero of $p\left(x\right)$
$\therefore$ Integral zeroes of the given polynomial are $-1,+1$ and $-2.$