| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Express as product with specific form |
| Difficulty | Standard +0.3 This is a standard C2 factor theorem question with routine steps: evaluate f(1) to find a factor, perform polynomial division to factorize completely, sketch a cubic with a repeated root, and integrate between roots. All techniques are textbook exercises requiring no novel insight, though the multi-part structure and integration at the end place it slightly above average difficulty. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f(1) = 1 - 9 + 24 - 16 = 0\) | B1 | |
| \(\therefore (x-1)\) is a factor of \(f(x)\) | B1 | |
| (b) Polynomial long division: | M1 A1 | |
| \(f(x) = (x-1)(x^2 - 8x + 16)\) | M1 A1 | |
| \(f(x) = (x-1)(x-4)^2\) | [\(p = -1, q = -4\)] | M1 A1 |
| (c) Sketch showing: curve passing through origin, local maximum between \(0\) and \(4\), touching \(x\)-axis at \(x = 4\), and increasing after. | B2 | |
| (d) \(= \int_1^4 (x^3 - 9x^2 + 24x - 16) \, dx\) | M1 | |
| \(= \left[\frac{1}{4}x^4 - 3x^3 + 12x^2 - 16x\right]_1^4\) | M1 A2 | |
| \(= [(64 - 192 + 192 - 64) - (-\frac{1}{4} - 3 + 12 - 16)]\) | M1 | |
| \(= 6\frac{1}{4}\) | A1 | (13 marks) |
**(a)** $f(1) = 1 - 9 + 24 - 16 = 0$ | B1 |
$\therefore (x-1)$ is a factor of $f(x)$ | B1 |
**(b)** Polynomial long division: | M1 A1 |
$f(x) = (x-1)(x^2 - 8x + 16)$ | M1 A1 |
$f(x) = (x-1)(x-4)^2$ | [$p = -1, q = -4$] | M1 A1 |
**(c)** Sketch showing: curve passing through origin, local maximum between $0$ and $4$, touching $x$-axis at $x = 4$, and increasing after. | B2 |
**(d)** $= \int_1^4 (x^3 - 9x^2 + 24x - 16) \, dx$ | M1 |
$= \left[\frac{1}{4}x^4 - 3x^3 + 12x^2 - 16x\right]_1^4$ | M1 A2 |
$= [(64 - 192 + 192 - 64) - (-\frac{1}{4} - 3 + 12 - 16)]$ | M1 |
$= 6\frac{1}{4}$ | A1 | **(13 marks)**
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**Total: 75 marks**9.
$$f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16 .$$
\begin{enumerate}[label=(\alph*)]
\item Evaluate $\mathrm { f } ( 1 )$ and hence state a linear factor of $\mathrm { f } ( x )$.
\item Show that $\mathrm { f } ( x )$ can be expressed in the form
$$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 } ,$$
where $p$ and $q$ are integers to be found.
\item Sketch the curve $y = \mathrm { f } ( x )$.
\item Using integration, find the area of the region enclosed by the curve $y = \mathrm { f } ( x )$ and the $x$-axis.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q9 [13]}}