| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2017 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Verify factor then factorise/solve |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing standard applications of remainder/factor theorem and factorisation. Parts (a)-(c) involve direct substitution and routine algebraic manipulation with clear guidance ('use', 'show that', 'hence'). Part (d) requires reading a graph to identify where the cubic is non-positive, which is a basic skill. All techniques are standard textbook exercises with no problem-solving insight required. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks |
|---|---|
| Attempts \(f(\pm 1)\) | M1 |
| Remainder \(= 2\) | A1 |
| Answer | Marks |
|---|---|
| Attempts \(f(\pm 3) = -4\times(\pm3)^3 + 16\times(\pm3)^2 - 13\times(\pm3) + 3\) | M1 |
| Remainder \(= 0 \Rightarrow (x-3)\) is a factor | A1* |
| Answer | Marks |
|---|---|
| Divides \(f(x)\) by \((x-3)\) to get quadratic factor \((-4x^2+4x-1)\) | M1, A1 |
| \(f(x) = (x-3)\times-(2x-1)(2x-1) = (3-x)(2x-1)^2\) | dM1 A1 |
| Answer | Marks |
|---|---|
| \(f(x) \leq 0 \Rightarrow (3-x)(2x-1)^2 \leq 0\) | |
| \(x = \frac{1}{2}\) | B1 |
| \(x \geq 3\) | B1 |
## Question 5:
**(a)**
| Attempts $f(\pm 1)$ | M1 | |
| Remainder $= 2$ | A1 | |
**(b)**
| Attempts $f(\pm 3) = -4\times(\pm3)^3 + 16\times(\pm3)^2 - 13\times(\pm3) + 3$ | M1 | |
| Remainder $= 0 \Rightarrow (x-3)$ is a factor | A1* | |
**(c)**
| Divides $f(x)$ by $(x-3)$ to get quadratic factor $(-4x^2+4x-1)$ | M1, A1 | |
| $f(x) = (x-3)\times-(2x-1)(2x-1) = (3-x)(2x-1)^2$ | dM1 A1 | |
**(d)**
| $f(x) \leq 0 \Rightarrow (3-x)(2x-1)^2 \leq 0$ | | |
| $x = \frac{1}{2}$ | B1 | |
| $x \geq 3$ | B1 | |5.
$$f ( x ) = - 4 x ^ { 3 } + 16 x ^ { 2 } - 13 x + 3$$
\begin{enumerate}[label=(\alph*)]
\item Use the remainder theorem to find the remainder when $\mathrm { f } ( x )$ is divided by ( $x - 1$ ).
\item Use the factor theorem to show that ( $x - 3$ ) is a factor of $\mathrm { f } ( x )$.
\item Hence fully factorise $\mathrm { f } ( x )$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-12_581_636_731_657}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$.
\item Use your answer to part (c) and the sketch to deduce the set of values of $x$ for which $\mathrm { f } ( x ) \leqslant 0$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2017 Q5 [10]}}