Question

The polynomial $f\left(x\right)=x^6+4x^5+x^4-12x^3-11x^2+4x+4$ has a zero of multiplicity $2$ at $x=-2$. Find the other real zeros.

Answer 1

If $f$ has a zero of multiplicity $2$, then it may be written as follows

$f\left(x\right)=(x+2{)}^{2}Q(x)$

Where $Q\left(x\right)$ is a polynomial of degree $4$ and may be found by division

$Q\left(x\right)=\frac{f\left(x\right)}{(x+2{)}^2}=x^4-3x^2+1$

Polynomial $f$ may now be written as

$f\left(x\right)=(x+2{)}^2(x^4-3x^2+1)$

The remaining zeros of polynomial f may be found by solving the equation

$x^4-3x^2+1=0$

It is an equation of the quadratic type with solutions

$\frac{(\sqrt{5}+1)}{2}$, $\frac{(\sqrt{5}-1)}{2}$ ,$\frac{(-\sqrt{5}-1)}{2}$ , $\frac{(-\sqrt{5}+1)}{2}$