NJC Complex Numbers Tutorial Q6
(i)
Prove that for any complex variable $z$ and complex number $\alpha $,
$\left( z-\alpha \right)\left( z-{{\alpha }^{*}} \right)={{z}^{2}}-\left( 2\operatorname{Re}\left( \alpha \right) \right)z+{{\left| \alpha \right|}^{2}}$.
(ii)
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.
(iii)
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.)
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