NJC Complex Numbers Tutorial Q6

Solved by

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


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.)

Suggested Video Solutions

Students Only

Login here to view
Join Us

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

Share with your friends!

Continue reading

Published: 5th March 2024

Leave a Reply

Your email address will not be published. Required fields are marked *