Quick questions on Complex number algebra explained: H2 Mathematics Vectors and Complex Numbers

When a real polynomial has a known complex root, the fastest way to extract a real factor is to use the sum and product of the conjugate pair, rather than expanding two complex linear factors. If $a + bi$ is a root, the pair has sum $2a$ and product $a^2 + b^2$, so the real quadratic factor is $z^2 - (2a)z + (a^2 + b^2)$. For the root $2 - 3i$, the factor is $z^2 - 4z + 13$. This shortcut turns "factor a quartic given one complex root" into a quick subtraction problem: divide the polynomial by this real quadratic to reveal the remaining factor, all without ever multiplying complex numbers together.

Express $(2 + i)(3 - 2i)$ in the form $a + bi$. [2 marks]

Given $1 + i$ is a root of a real quadratic $z^2 + bz + c = 0$, find $b$ and $c$. [3 marks]