| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a standard Further Maths question on complex roots with real coefficients. Part (a) requires recalling that complex roots come in conjugate pairs (routine knowledge). Part (b) involves forming a quadratic factor from the conjugate pair, polynomial division, and solving the resulting quadratic—all standard techniques. Part (c) is straightforward plotting. While it's a multi-part question requiring several steps, each step follows a well-established procedure with no novel insight needed. It's slightly easier than average even for Further Maths because the method is highly algorithmic. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots |
8.
$$f ( z ) = z ^ { 4 } + 6 z ^ { 3 } + 76 z ^ { 2 } + a z + b$$
where $a$ and $b$ are real constants.
Given that $- 3 + 8 \mathrm { i }$ is a complex root of the equation $\mathrm { f } ( \mathrm { z } ) = 0$
\begin{enumerate}[label=(\alph*)]
\item write down another complex root of this equation.
\item Hence, or otherwise, find the other roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$
\item Show on a single Argand diagram all four roots of the equation $f ( z ) = 0$
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2016 Q8 [9]}}