| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Verify factor then sketch or analyse curve |
| Difficulty | Moderate -0.8 This is a straightforward C1 question requiring routine application of the factor theorem (substitute x=2 to verify), polynomial division to find the quadratic factor, factorising a simple quadratic, then sketching a cubic with given roots. All steps are standard textbook procedures with no problem-solving insight required, making it easier than average but not trivial due to the multi-step nature. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| \(p(2) = 8 + 4 - 20 + 8 = 0, \Rightarrow x - 2\) is a factor | M1 | Finding p(2); M0 long division |
| A1 | 2 | Shown \(= 0\) AND conclusion/statement about x – 2 being a factor |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt at quadratic factor | M1 | or factor theorem again for 2nd factor or (x+4) or (x–1) proved to be a factor |
| \(x^2 + 3x - 4\) | A1 | — |
| \(p(x) = (x-2)(x+4)(x-1)\) | A1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Graph through (0,8) | B1 | 8 marked |
| Ft "their factors" | B1√ | — |
| Cubic curve through their 3 points | M1 | — |
| Correct including x-intercepts correct | A1 | 4 |
## 6(a)(i)
$p(2) = 8 + 4 - 20 + 8 = 0, \Rightarrow x - 2$ is a factor | M1 | Finding p(2); M0 long division
| A1 | 2 | Shown $= 0$ **AND** conclusion/statement about x – 2 being a factor
## 6(a)(ii)
Attempt at quadratic factor | M1 | or factor theorem again for 2nd factor or (x+4) or (x–1) proved to be a factor
$x^2 + 3x - 4$ | A1 | —
$p(x) = (x-2)(x+4)(x-1)$ | A1 | 3 | or $(x+4)$ or $(x-1)$ proved to be a factor
## 6(b)
Graph through (0,8) | B1 | 8 marked
Ft "their factors" | B1√ | — | 3 roots marked on x-axis
Cubic curve through their 3 points | M1 | —
Correct including x-intercepts correct | A1 | 4 | Condone max on y-axis etc or slightly wrong concavity at ends of graph
---6 The polynomial $\mathrm { p } ( x )$ is given by
$$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 10 x + 8$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Using the factor theorem, show that $x - 2$ is a factor of $\mathrm { p } ( x )$.
\item Hence express $\mathrm { p } ( x )$ as the product of three linear factors.
\end{enumerate}\item Sketch the curve with equation $y = x ^ { 3 } + x ^ { 2 } - 10 x + 8$, showing the coordinates of the points where the curve cuts the axes.\\
(You are not required to calculate the coordinates of the stationary points.)
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2006 Q6 [9]}}