| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2023 |
| Session | February |
| Marks | 5 |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a straightforward Further Maths complex roots question requiring knowledge that complex roots come in conjugate pairs for real polynomials. Part (a) involves routine polynomial division or reconstruction of quadratic factors from known roots, and part (b) is immediate once (a) is complete. The 5-mark allocation and explicit guidance make this easier than average, though it does require Further Maths content knowledge. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem |
In this question you must show detailed reasoning.
The equation f(x) = 0, where f(x) = $x^4 + 2x^3 + 2x^2 + 26x + 169$, has a root x = 2 + 3i.
\begin{enumerate}[label=(\alph*)]
\item Express f(x) as a product of two quadratic factors. [4]
\item Hence write down all the roots of the equation f(x) = 0. [1]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2023 Q8 [5]}}