| 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 straightforward multi-part question testing the Remainder Theorem and factorization. Part (a) is direct substitution, part (b) is routine algebraic manipulation or polynomial division, and part (c) requires recognizing that consecutive integers guarantee divisibility by 3. While it requires multiple steps, each technique is standard C2 material with clear signposting, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
$\text{f}(n) = n^3 + pn^2 + 11n + 9$, 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 Q3 [7]}}