Perform complex-number arithmetic: addition, subtraction, multiplication and division, with and without technology.

Given the complex numbers $z_1 = 3 + 4i$ and $z_2 = 1 - 2i$, calculate $z_1 \times z_2$.

Calculate \(z_1 \times z_2\).

Let \(z_1 = 3 + 2i\) and \(z_2 = 1 - 4i\). a) Determine \(z_1 + z_2\) and express your answer in the form \(a + bi\). (1 mark) b) Determine \(z_1 \times z_2\) and express your answer in the form \(a + bi\). (1 mark) c) Determine \(\frac{z_1}{z_2}\) and express your answer in the form \(a + bi\), where \(a\) and \(b\) are exact rational values. (1 mark)

A communications engineer models a signal transformation using complex numbers. The input signal is represented by \(z_1 = 3 + 4i\) and passes through a filter represented by \(z_2 = 2 - i\). a) Determine the product \(z_1 \times z_2\) in the form \(a + bi\), where \(a, b \in \mathbb{R}\). Show your working. b) Use your result from part (a) to determine the quotient \(\frac{z_1 \times z_2}{z_2}\) without technology, simplifying your answer to the form \(a + bi\). c) Verify your result from part (b) algebraically.