| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | February |
| Marks | 6 |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a standard Further Maths complex roots question requiring knowledge that complex roots come in conjugate pairs for real coefficient polynomials. Students must find the conjugate root (1+3i), form the quadratic factor (z-(1-3i))(z-(1+3i)) = z²-2z+10, then divide the quartic to find the other quadratic factor. While it requires multiple steps and careful algebra, it follows a well-practiced procedure with no novel insight needed. Slightly easier than average due to the structured guidance. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem |
It is given that $1 - 3i$ is one root of the quartic equation
$$z^4 - 2z^3 + pz^2 + rz + 80 = 0$$
where $p$ and $r$ are real numbers.
\begin{enumerate}[label=(\alph*)]
\item Express $z^4 - 2z^3 + pz^2 + rz + 80$ as the product of two quadratic factors with real coefficients. [4 marks]
\item Find the value of $p$ and the value of $r$. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q3 [6]}}