Standard +0.3 This is a slightly-above-average A-level question requiring knowledge that complex roots come in conjugate pairs for polynomials with real coefficients, then finding the quadratic factor and solving. The steps are systematic: use conjugate root theorem, expand (z-(3+2i))(z-(3-2i)) to get z²-6z+13, divide the quartic by this quadratic, solve the resulting quadratic, and plot four points on an Argand diagram. While it requires multiple techniques, each step follows standard procedures taught in Core Pure 1 with no novel insight needed.
3.
$$\mathrm { f } ( z ) = z ^ { 4 } + a z ^ { 3 } + 6 z ^ { 2 } + b z + 65$$
where \(a\) and \(b\) are real constants.
Given that \(z = 3 + 2 \mathbf { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\), show the roots of \(\mathrm { f } ( z ) = 0\) on a single Argand diagram.
3.
$$\mathrm { f } ( z ) = z ^ { 4 } + a z ^ { 3 } + 6 z ^ { 2 } + b z + 65$$
where $a$ and $b$ are real constants.\\
Given that $z = 3 + 2 \mathbf { i }$ is a root of the equation $\mathrm { f } ( z ) = 0$, show the roots of $\mathrm { f } ( z ) = 0$ on a single Argand diagram.
\hfill \mbox{\textit{Edexcel CP1 Q3 [9]}}