| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 7 |
| Topic | Factor & Remainder Theorem |
| Type | Known polynomial, verify then factorise |
| Difficulty | Moderate -0.8 This is a straightforward, routine application of the factor theorem with standard steps: verify a given factor by substitution, perform polynomial division to find the quadratic factor, then solve using the quadratic formula. All techniques are basic and the question provides significant scaffolding by giving the linear factor upfront. Easier than average A-level content. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
1.
$$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$
\begin{enumerate}[label=(\alph*)]
\item Use the factor theorem to show that $( x + 3 )$ is a factor of $f ( x )$.\\
(2)
\item Factorise $f ( x )$ to a linear and quadratic factor.\\
(2)
\item Hence find exact values for all the solutions of the equation $\mathrm { f } ( x ) = 0$\\
(3)
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q1 [7]}}