| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| 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 immediate recall that complex roots come in conjugate pairs. Part (b) involves forming a quadratic factor from the conjugate pair, polynomial division, and solving the resulting quadratic. Part (c) uses coefficient comparison. While it requires multiple techniques, each step follows a well-practiced procedure with no novel insight needed. Slightly easier than average due to its routine nature. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem |
4.
$$f ( z ) = 2 z ^ { 4 } - 19 z ^ { 3 } + A z ^ { 2 } + B z - 156$$
where $A$ and $B$ are constants.\\
The complex number $5 - \mathrm { i }$ is a root of the equation $\mathrm { f } ( \mathrm { z } ) = 0$
\begin{enumerate}[label=(\alph*)]
\item Write down another complex root of this equation.
\item Solve the equation $\mathrm { f } ( \mathrm { z } ) = 0$ completely.
\item Determine the value of $A$ and the value of $B$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2022 Q4 [8]}}