Question

In the following polynomial, its one of zero is -2, find all other zeroes.
$f\left(x\right)=x^3+13x^2+32x+20$

Answer 1

Given
$f\left(x\right)=x^3+13x^2+32x+20$ and one zero is $-2$
$(x+2)$ will be factor of $f\left(x\right)$
Now, dividing $f\left(x\right)$ by $x+2$
$x^2+11x+10x+2)\overline{x^3+13x^2+32x+2\_0}{x}^3+2x^2--\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_11x^2+32x+2011x^2+22x--\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_10x+2010x+20--\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_0$
Thus $f\left(x\right)=(x+2)(x^2+11x+10)$
$=(x+2)[x^2+10x+x+10]$
$=(x+2)\left[x\right(x+10)+1(x+10\left)\right]$
$=(x+2)(x+10)(x+1)$
Nowe, other zeroes of polynomial
If $x+10=0$ then $x=-10$
or $x+1=0$ then $x=-1$
Thus, other zeroes of polynomial are $-10,-1$