| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Moderate -0.3 This is a straightforward application of the complex conjugate root theorem and polynomial division. Given one complex root of a polynomial with real coefficients, students immediately know its conjugate is also a root, then find the third root via factorization or Vieta's formulas. This is a standard FP1 exercise requiring routine technique rather than problem-solving insight, making it slightly easier than average. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02h Square roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(1 - 2i\) (is also a root) | B1 | seen |
| \((z-(1+2i))(z-(1-2i)) = z^2 - 2z + 5\) | M1A1 | Attempt to expand \((z-(1+2i))(z-(1-2i))\) or any valid method to establish the quadratic factor |
| \(f(z) = (z^2 - 2z + 5)(2z + 1)\) | M1 | Attempt at linear factor with their \(cd\) in \((z^2 + az + c)(2z + d) = \pm 5\) or \((z^2 - 2z + 5)(2z + a) \Rightarrow 5a = 5\) |
| \(z_3 = -\frac{1}{2}\) | A1 |
## Question 1:
$f(z) = 2z^3 - 3z^2 + 8z + 5$
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1 - 2i$ (is also a root) | B1 | seen |
| $(z-(1+2i))(z-(1-2i)) = z^2 - 2z + 5$ | M1A1 | Attempt to expand $(z-(1+2i))(z-(1-2i))$ or any valid method to establish the quadratic factor |
| $f(z) = (z^2 - 2z + 5)(2z + 1)$ | M1 | Attempt at **linear** factor with their $cd$ in $(z^2 + az + c)(2z + d) = \pm 5$ or $(z^2 - 2z + 5)(2z + a) \Rightarrow 5a = 5$ |
| $z_3 = -\frac{1}{2}$ | A1 | |
---\begin{enumerate}
\item The roots of the equation
\end{enumerate}
$$2 z ^ { 3 } - 3 z ^ { 2 } + 8 z + 5 = 0$$
are $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$\\
Given that $z _ { 1 } = 1 + 2 i$, find $z _ { 2 }$ and $z _ { 3 }$\\
\hfill \mbox{\textit{Edexcel FP1 2014 Q1 [5]}}