| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Non-zero remainder condition |
| Difficulty | Standard +0.3 This is a structured multi-part question on the remainder theorem and factor theorem with clear scaffolding. Part (a) is direct application of the remainder theorem, part (b) follows naturally once p is known, and part (c) requires recognizing that consecutive integers guarantee divisibility by 3. While it requires multiple techniques, the question guides students through each step with no novel insight needed, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
$f(n) = n^3 + pn^2 + 11n + 9$, where $p$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Given that f(n) has a remainder of 3 when it is divided by $(n + 2)$, prove that $p = 6$. [2]
\item Show that f(n) can be written in the form $(n + 2)(n + q)(n + r) + 3$, where $q$ and $r$ are integers to be found. [3]
\item Hence show that f(n) is divisible by 3 for all positive integer values of $n$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q2 [7]}}