| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Known polynomial, verify then factorise |
| Difficulty | Easy -1.2 This is a straightforward C1 question testing routine application of Factor and Remainder Theorems with standard algebraic manipulation. Part (a) requires simple substitution, (b) involves factoring out x then factorising a quadratic, and (c) applies textbook methods with no problem-solving insight needed. All steps are mechanical and well-practiced at this level. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
6 The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 3 x$.
\begin{enumerate}[label=(\alph*)]
\item Use the Factor Theorem to show that $x - 3$ is a factor of $\mathrm { p } ( x )$.
\item Express $\mathrm { p } ( x )$ as the product of three linear factors.
\item \begin{enumerate}[label=(\roman*)]
\item Use the Remainder Theorem to find the remainder, $r$, when $\mathrm { p } ( x )$ is divided by $x - 2$.
\item Using algebraic division, or otherwise, express $\mathrm { p } ( x )$ in the form
$$( x - 2 ) \left( x ^ { 2 } + a x + b \right) + r$$
where $a , b$ and $r$ are constants.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2006 Q6 [10]}}