Practice section 3 of 9

How do we rationalize $\dfrac{1}{\sqrt a+\sqrt b}$?

By expressing $z=\dfrac{5-3i}{2+3i}$ in the form $a+ib$, find the real and imaginary parts of $z$ in exact form.

(a) Find $z_1/z_2$ for each pair:

(i) $z_1=2+3i$, $z_2=3-4i$;

(ii) $z_1=3-i$, $z_2=4-i$;

(iii) $z_1=-2+5i$, $z_2=1-3i$.

(b) Express the following in the form $a+ib$, where $a$ and $b$ are exact real numbers:

(i) $\dfrac1{1+3i}+\dfrac1{1-3i}$;

(ii) $-2+i+\dfrac{3i}{2-i}$;

(iii) $\dfrac{1-2i}{3+4i}+\dfrac{-2-i}{4+3i}$.

A graphing calculator is not to be used in answering this question.

Solve the simultaneous equations $z+(i-1)w=0$ and $3z+(1-i)w=1$.

Given that $2+i$ is a root of the equation $z^3+az^2+z+b=0$, find the values of the real constants $a$ and $b$. Hence, find the other roots.

Verify that $z=2i$ is a root of the equation $z^4-2z^3+7z^2-8z+12=0$. Hence, determine the other roots.