Question

Find all the zeroes of the polynomial $\left(x\right)=x^3+13x^2+32x+20,$ if one of its zeroes is -2.

Answer 1

Polynomial is $p\left(x\right)=x^3+13x^2+32x+20$

one of the zero is $x=-2$

One factor of $p\left(x\right)$ is $(x+2)$

$\Rightarrow$ Polynomial becomes $p\left(x\right)=(x+2)(x^2+11x+10)$ factoring the quadratic, by middle term spletting

$\Rightarrow$ $p\left(x\right)=(x+2)(x^2+10x+x+10)$

$=(x+2)\left[x\right(x+10)+1(x+10\left)\right]$

$=(x+2)\left[\right(x+10\left)\right(x+1\left)\right]$

$\Rightarrow$ Other factor are $x+10$ and $x+1$

Zeros are $x=-10$ and $x=-1$

$\Rightarrow$ All the zeroes of polynomial $p\left(x\right)$ are $-1,-2$ and $-10$