| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2023 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Factor theorem and finding roots |
| Difficulty | Standard +0.3 This is a straightforward Further Maths F1 question on polynomial roots requiring standard techniques: substitution to find p, polynomial division/factorization, solving a quadratic (likely complex roots), finding modulus, and plotting on Argand diagram. All steps are routine for FM students with no novel insight required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02g Conjugate pairs: real coefficient polynomials4.02h Square roots: of complex numbers4.02i Quadratic equations: with complex roots4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation |
\begin{enumerate}
\item In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
\end{enumerate}
$$\mathrm { f } ( z ) = 4 z ^ { 3 } + p z ^ { 2 } - 24 z + 108$$
where $p$ is a constant.\\
Given that - 3 is a root of the equation $\mathrm { f } ( \mathrm { z } ) = 0$\\
(a) determine the value of $p$\\
(b) using algebra, solve $\mathrm { f } ( \mathrm { z } ) = 0$ completely, giving the roots in simplest form,\\
(c) determine the modulus of the complex roots of $\mathrm { f } ( \mathrm { z } ) = 0$\\
(d) show the roots of $\mathrm { f } ( \mathrm { z } ) = 0$ on a single Argand diagram.
\hfill \mbox{\textit{Edexcel F1 2023 Q3 [10]}}