NJC Complex Numbers Tutorial Q6

Prove that for any complex variable \(z\) and complex number \(\alpha \),

\(\left( z-\alpha \right)\left( z-{{\alpha }^{*}} \right)={{z}^{2}}-\left( 2\operatorname{Re}\left( \alpha \right) \right)z+{{\left| \alpha \right|}^{2}}\).

Explain why the argument \(\theta \) of any root \(\alpha \) of the equation \({{z}^{5}}+32=0\) must be of the form \(\frac{2k+1}{5}\pi \) for some integer \(k\) and find the common modulus of all the roots of this equation.

Use your results in parts (i) and (ii) to express \({{z}^{5}}+32\) as a product of one linear factor and two quadratic factors, all with real coefficients.

(You may express the coefficients in trigonometric form, where necessary.)