How do I solve this complex polynomial?

Question

I'm a bit stuck on this question, it asks to solve $P(z)=0$ over $\Bbb{C}$ and factorise $P(z)$ fully over $\Bbb{R}$. $$P(z)=5z^4-11z^3+16z^2-11z+5$$ Am I supposed to solve this by using the substitution method (i.e. $z=y-\frac{b} a$)? If so, where do I go from there?

Any help is greatly appreciated!!

Answer 1

Equations of this type where the coefficients located at equal distances from the middle one are equal can be solved in a standard way. First, notice that $z=0$ is not a root, so you can divide by $z^2$ and then you can group as follows: $$5z^2-11z+16-\frac{11}{z}+\frac{5}{z^2}=0$$ $$5\left(z^2+\frac{1}{z^2}\right)-11\left(z+\frac{1}{z}\right)+16=0$$ Then, let $y=z+\frac{1}{z}$ and notice $z^2+\frac{1}{z^2}=y^2-2$. This transforms your equation into a quadratic one.

Answer 2

Hint: it factors as the product of two quadratics with integer coefficients, both of which have the same symmetry.