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.)
Suggested Video Solutions
Our H2 Math Tuition includes
- Question Bank with Video solutions to 1400+ questions
- Online Portal
- H2 Math Summary Notes
- Structured Curriculum and Notes
Share with your friends!
Continue reading
5 Exam Preparation Tips: How We Help Students Excel in Exams
In the ever-evolving landscape of education, where academic success is often measured by performance in exams, the need for effective test preparation has never been more critical. Students face a
List MF 27
What is the MF 27? The MF 27, set to replace the MF 26 from 2025, is a comprehensive formula sheet developed by the Ministry of Education Singapore in collaboration
Panic No More: What To Do The Day Before An Exam?
As you approach the final hours before your exam, remember that a balanced approach is key. A light review can keep the information fresh in your mind, while preparing your