| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Verify factor then sketch or analyse curve |
| Difficulty | Moderate -0.3 This is a standard C2 factor theorem question with routine steps: verify a factor by substitution, perform polynomial division to fully factorise, identify a repeated root as a turning point, and use basic differentiation. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f(3) = 27 - 36 - 9 + 18 = 0 \therefore (x-3)\) is a factor | M1 A1 | |
| (b) Long division: | M1 A1 | |
| \(f(x) = (x-3)(x^2 - x - 6)\) | ||
| \(f(x) = (x-3)(x+2)(x-3) = (x+2)(x-3)^2\) | M1 A1 | |
| (c) \((3, 0)\) | B1 | |
| \((x-3)\) is a repeated factor of \(f(x) \therefore\) x-axis is tangent where \(x = 3\) | B1 | |
| (d) \(f'(x) = 3x^2 - 8x - 3\) | M1 A1 | |
| for SP, \(3x^2 - 8x - 3 = 0\) | M1 | |
| \((3x+1)(x-3) = 0\) | M1 | |
| \(x = -\frac{1}{3}, 3\) | A1 | (13 marks) |
(a) $f(3) = 27 - 36 - 9 + 18 = 0 \therefore (x-3)$ is a factor | M1 A1 |
(b) Long division: | M1 A1 |
$f(x) = (x-3)(x^2 - x - 6)$ |
$f(x) = (x-3)(x+2)(x-3) = (x+2)(x-3)^2$ | M1 A1 |
(c) $(3, 0)$ | B1 |
$(x-3)$ is a repeated factor of $f(x) \therefore$ x-axis is tangent where $x = 3$ | B1 |
(d) $f'(x) = 3x^2 - 8x - 3$ | M1 A1 |
for SP, $3x^2 - 8x - 3 = 0$ | M1 |
$(3x+1)(x-3) = 0$ | M1 |
$x = -\frac{1}{3}, 3$ | A1 | (13 marks)
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**Total: 75 marks**9. $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18$.
\begin{enumerate}[label=(\alph*)]
\item Show that $( x - 3 )$ is a factor of $\mathrm { f } ( x )$.
\item Fully factorise $\mathrm { f } ( x )$.
\item Using your answer to part (b), write down the coordinates of one of the turning points of the curve $y = \mathrm { f } ( x )$ and give a reason for your answer.
\item Using differentiation, find the $x$-coordinate of the other turning point of the curve $y = \mathrm { f } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q9 [13]}}