| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2022 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.8 This is a structured Further Maths question on polynomial roots requiring knowledge of complex conjugate pairs and use of the constant term to find the repeated root. While multi-step, each part guides students through standard techniques (conjugate pairs, forming quadratic factors, using product of roots). The novel element is recognizing that the constant term 225 = 25 × 9 helps identify the repeated root as 5, but the scaffolding makes this accessible for FM students. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots4.05a Roots and coefficients: symmetric functions |
The equation
$$x^4 + Ax^3 + Bx^2 + Cx + 225 = 0$$
where $A$, $B$ and $C$ are real constants, has
\begin{itemize}
\item a complex root $4 + 3\text{i}$
\item a repeated positive real root
\end{itemize}
\begin{enumerate}[label=(\alph*)]
\item Write down the other complex root of this equation.
[1]
\item Hence determine a quadratic factor of $x^4 + Ax^3 + Bx^2 + Cx + 225$
[2]
\item Deduce the real root of the equation.
[2]
\item Hence determine the value of each of the constants $A$, $B$ and $C$
[3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2022 Q4 [8]}}