| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Factor theorem and finding roots |
| Difficulty | Moderate -0.8 This is a straightforward Further Pure question requiring polynomial division to find coefficients, then solving a quadratic (which has complex roots). Part (c) is trivial using Vieta's formulas or summing the found roots. All steps are routine techniques with no problem-solving insight required, making it easier than average despite being FP1. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(a = -2, b = 50\) | B1, B1 | (2 marks) |
| (b) \(-3\) is a root | B1 | |
| Solving 3-term quadratic: \(x = \frac{2 \pm \sqrt{4 - 200}}{2}\) or \((x-1)^2 - 1 + 50 = 0\) | M1 | |
| \(= 1 + 7i, 1 - 7i\) | A1, A1 ft | A1: for a correct root (simplified as here) and A1 ft: for conjugate of first answer. Accept correct answers with no working here. If answers are written down as factors then isw. Must see roots for marks. |
| (c) \((-3) + (1 + 7i) + (1 - 7i) = -1\) | B1 ft | Requires the sum of two non-real conjugate roots and a real root resulting in a real number. Answers including \(x\) are B0 |
**(a)** $a = -2, b = 50$ | **B1, B1** | (2 marks)
**(b)** $-3$ is a root | **B1** |
Solving 3-term quadratic: $x = \frac{2 \pm \sqrt{4 - 200}}{2}$ or $(x-1)^2 - 1 + 50 = 0$ | **M1** |
$= 1 + 7i, 1 - 7i$ | **A1, A1 ft** | A1: for a correct root (simplified as here) and A1 ft: for conjugate of first answer. Accept correct answers with no working here. If answers are written down as factors then isw. Must see roots for marks.
**(c)** $(-3) + (1 + 7i) + (1 - 7i) = -1$ | **B1 ft** | Requires the sum of two non-real conjugate roots and a real root resulting in a real number. Answers including $x$ are **B0**
**Total: 7 marks**
---4.
$$f ( x ) = x ^ { 3 } + x ^ { 2 } + 44 x + 150$$
Given that $\mathrm { f } ( x ) = ( x + 3 ) \left( x ^ { 2 } + a x + b \right)$, where $a$ and $b$ are real constants,
\begin{enumerate}[label=(\alph*)]
\item find the value of $a$ and the value of $b$.
\item Find the three roots of $\mathrm { f } ( x ) = 0$.
\item Find the sum of the three roots of $\mathrm { f } ( x ) = 0$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2010 Q4 [7]}}