| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Moderate -0.3 This is a standard Further Maths question testing the fundamental property that complex roots come in conjugate pairs for polynomials with real coefficients. Part (a) requires immediate recall, part (b) involves routine expansion of factors and polynomial division, and part (c) uses coefficient comparison. While it's Further Maths content, it follows a completely predictable template with no problem-solving insight required, making it slightly easier than an average A-level question overall. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(1 + 2i\) | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \((x-1+2i)(x-1-2i)\) are factors of \(f(x)\) | M1 | |
| So \(x^2 - 2x + 5\) is a factor of \(f(x)\) | M1 A1 | |
| \(f(x) = (x^2-2x+5)(2x-1)\) | M1 A1ft | |
| Third root is \(\frac{1}{2}\) | A1 | (6 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(p = 10 + 2\) | M1 | |
| \(= 12\) | A1 | (2 marks) |
# Question 8:
## Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $1 + 2i$ | B1 | **(1 mark)** |
## Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $(x-1+2i)(x-1-2i)$ are factors of $f(x)$ | M1 | |
| So $x^2 - 2x + 5$ is a factor of $f(x)$ | M1 A1 | |
| $f(x) = (x^2-2x+5)(2x-1)$ | M1 A1ft | |
| Third root is $\frac{1}{2}$ | A1 | **(6 marks)** |
## Part (c):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $p = 10 + 2$ | M1 | |
| $= 12$ | A1 | **(2 marks)** |
---8.
$$\mathrm { f } ( x ) \equiv 2 x ^ { 3 } - 5 x ^ { 2 } + p x - 5 , p \in \mathbb { R }$$
Given that $1 - 2 \mathrm { i }$ is a complex solution of $\mathrm { f } ( x ) = 0$,
\begin{enumerate}[label=(\alph*)]
\item write down the other complex solution of $\mathrm { f } ( x ) = 0$,
\item solve the equation $\mathrm { f } ( x ) = 0$,
\item find the value of $p$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q8 [9]}}