| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Sketch curve using polynomial roots |
| Difficulty | Moderate -0.8 This is a standard C1 Factor Theorem question with routine steps: substitute x=-2 to find k, factorise by inspection/division, apply Remainder Theorem, and sketch a cubic. All techniques are textbook exercises requiring minimal problem-solving, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Must have statement if \(k=10\) substitute | M1, A1 | or long division or \((x+2)(x^2 - 6x + 5)\); AG likely withheld if \(p(-2) = 0\) not seen |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow p(x) = (x + 2)(x - 1)(x - 5)\) | M1, A1, A1 | Attempt at quadratic or second linear factor attempt; Must be written as product |
| Answer | Marks | Guidance |
|---|---|---|
| Condone \(k - 30\) | M1, A1 | long division scores M0; Condone \(k - 30\) |
| Answer | Marks | Guidance |
|---|---|---|
| [Graph showing cubic with maximum at \((0, 10)\), crossing x-axis at \(x = -2, 1, 5\), with curve behavior marked] | B1, B1✓, M1, A1 | Curve thro' 10 marked on y-axis; FT their 3 roots marked on x-axis; Cubic shape with a max and min; Correct graph (roughly as on left) going beyond \(-2\) and \(5\) (condone max anywhere between \(x = -2\) and \(1\) and min between \(1\) and \(5\)) |
## 1(a)(i)
$p(-2) = -8 -16 +14 + k$
$p(-2) = 0 \Rightarrow -10 + k = 0 \Rightarrow k = 10$
Must have statement if $k=10$ substitute | M1, A1 | or long division or $(x+2)(x^2 - 6x + 5)$; AG likely withheld if $p(-2) = 0$ not seen
## 1(a)(ii)
$p(x) = (x + 2)(x^2 + \ldots 5)$
$p(x) = (x + 2)(x^2 - 6x + 5)$
$\Rightarrow p(x) = (x + 2)(x - 1)(x - 5)$ | M1, A1, A1 | Attempt at quadratic or second linear factor attempt; Must be written as product
## 1(b)
$p(3) = 27 -36 -21 + k$
(Remainder) $= k - 30 = -20$
Condone $k - 30$ | M1, A1 | long division scores M0; Condone $k - 30$
## 1(c)
[Graph showing cubic with maximum at $(0, 10)$, crossing x-axis at $x = -2, 1, 5$, with curve behavior marked] | B1, B1✓, M1, A1 | Curve thro' 10 marked on y-axis; FT their 3 roots marked on x-axis; Cubic shape with a max and min; Correct graph (roughly as on left) going beyond $-2$ and $5$ (condone max anywhere between $x = -2$ and $1$ and min between $1$ and $5$)
**Total: 11**
---1 The polynomial $\mathrm { p } ( x )$ is given by
$$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 7 x + k$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Given that $x + 2$ is a factor of $\mathrm { p } ( x )$, show that $k = 10$.
\item Express $\mathrm { p } ( x )$ as the product of three linear factors.
\end{enumerate}\item Use the Remainder Theorem to find the remainder when $\mathrm { p } ( x )$ is divided by $x - 3$.
\item Sketch the curve with equation $y = x ^ { 3 } - 4 x ^ { 2 } - 7 x + 10$, indicating the values where the curve crosses the $x$-axis and the $y$-axis. (You are not required to find the coordinates of the stationary points.)
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2007 Q1 [11]}}