# NJC Complex Numbers Tutorial Q6

## Timothy Gan

This is Tim. Tim loves to teach math. Tim seeks to improve his teaching incessantly! Help Tim by telling him how he can do better.
##### 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.)

## Students Only

Our H2 Math Tuition includes

• Question Bank with Video solutions to 1400+ questions
• Online Portal
• H2 Math Summary Notes
• Structured Curriculum and Notes
Free Stuff
Pure Math
Statistics

##### Goal Setting and Progress Tracking for Math Students: A Comprehensive Guide

In the pursuit of academic success, setting and achieving goals play a pivotal role, particularly in disciplines like mathematics. Effective goal setting not only provides students with a roadmap for