Complex Numbers

Given that $\displaystyle z_1 = 5 + 3i$ and $\displaystyle z_2 = 6 + 2i$, express $\displaystyle z_1\overline{z_2}$ in the form $\displaystyle a + ib$ where $\displaystyle a$ and $\displaystyle b$ are real numbers.

(a)Express $\displaystyle z = 1 + \sqrt{3}i$ in polar form.

(b)Hence, or otherwise, show that $\displaystyle z^3$ is real.

A complex number is defined by $\displaystyle z = 1 + i.$

(a)Express $\displaystyle z$ in polar form.

(b)Use de Moivre's theorem to evaluate $\displaystyle z^8.$

Given $\displaystyle z = x + iy$, $\displaystyle y \neq 0$, solve the equation

$\displaystyle z^2 + 20\bar{z} - 156 = 0$

where $\displaystyle \bar{z}$ is the complex conjugate of $\displaystyle z.$

Two complex numbers are defined as $\displaystyle z = 11 + 10i$ and $\displaystyle w = 3 - 2i.$
Find $\displaystyle \frac{z}{w}$ in the form $\displaystyle a + bi$, where $\displaystyle a, b \in \mathbb{R}.$

A complex number is defined by $\displaystyle z=\sqrt{3}+i.$

(a)Express $\displaystyle z$ in polar form.

(b)Use de Moivre's theorem to show that $\displaystyle z^{3}$ is purely imaginary.

The complex number $\displaystyle z=2+i$ is a root of the polynomial equation

$\displaystyle z^{4}-2z^{3}-z^{2}+2z+10=0.$

(a)State a second root of the equation.

(b)Find the remaining roots.