| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question testing standard knowledge that complex roots come in conjugate pairs for polynomials with real coefficients. Part (a) requires immediate recall, part (b)(i) involves routine polynomial division or coefficient comparison, and part (b)(ii) is direct substitution. While it's Further Maths content, the execution is mechanical with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem |
7.
$$f ( z ) = z ^ { 4 } + 4 z ^ { 3 } + 6 z ^ { 2 } + 4 z + a$$
where $a$ is a real constant.
Given that $1 + 2 \mathrm { i }$ 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 \begin{enumerate}[label=(\roman*)]
\item Hence, find the other roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$
\item State the value of $a$.
\includegraphics[max width=\textwidth, alt={}, center]{cfeb435a-03c2-4bcd-9c9f-6f62b4556cb3-15_31_33_205_2014}\\
" "\\
\includegraphics[max width=\textwidth, alt={}, center]{cfeb435a-03c2-4bcd-9c9f-6f62b4556cb3-15_42_53_317_1768}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2017 Q7 [8]}}