| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Sketch curve using polynomial roots |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing standard application of Remainder and Factor Theorems with routine polynomial factorisation and sketching. All parts follow textbook procedures with no problem-solving required—substitute values, perform algebraic division, identify roots, and sketch—making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| \(p(-1) = (-1)^3 - 4 \times (-1)^2 - 3(-1) + 18 = (= -1 - 4 + 3 + 18) = 16\) | M1, A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(p(3) = 3^3 - 4 \times 3^2 - 3 \times 3 + 18\) | M1 | \(p(3)\) attempted not long division |
| \(p(3) = 27 - 36 - 9 + 18 = 0 \Rightarrow x - 3\) is a factor | A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Quadratic factor \((x^2 - x + c)\) or \((x^2 + bx - 6)\) | M1 | \(-x\) or \(-6\) term by inspection or full long division by \(x - 3\) or comparing coefficients or \(p(-2)\) attempted; correct quadratic factor (or \(x+2\) shown to be factor by Factor Theorem) |
| \([p(x) =] (x - 3)(x - 3)(x + 2)\) | A1 | or \([p(x) =] (x - 3)^2(x + 2)\); must see product of factors |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | cubic curve with one maximum and one minimum | |
| A1 | meeting \(x\)-axis at \(-2\) and touching \(x\)-axis at \(3\) | |
| A1 | 3 marks | graph as shown, going beyond \(x = -2\) but condone max on or to right of \(y\)-axis |
**5(a)**
$p(-1) = (-1)^3 - 4 \times (-1)^2 - 3(-1) + 18 = (= -1 - 4 + 3 + 18) = 16$ | M1, A1 | 2 marks | $p(-1)$ attempted not long division
**5(b)(i)**
$p(3) = 3^3 - 4 \times 3^2 - 3 \times 3 + 18$ | M1 | $p(3)$ attempted not long division
$p(3) = 27 - 36 - 9 + 18 = 0 \Rightarrow x - 3$ is a factor | A1 | 2 marks | shown $= 0$ plus statement
**5(b)(ii)**
Quadratic factor $(x^2 - x + c)$ or $(x^2 + bx - 6)$ | M1 | $-x$ or $-6$ term by inspection or full long division by $x - 3$ or comparing coefficients or $p(-2)$ attempted; correct quadratic factor (or $x+2$ shown to be factor by Factor Theorem)
$[p(x) =] (x - 3)(x - 3)(x + 2)$ | A1 | or $[p(x) =] (x - 3)^2(x + 2)$; must see product of factors
**5(c)**
| M1 | cubic curve with one maximum and one minimum
| A1 | meeting $x$-axis at $-2$ and touching $x$-axis at $3$
| A1 | 3 marks | graph as shown, going beyond $x = -2$ but condone max on or to right of $y$-axis
---5 The polynomial $\mathrm { p } ( x )$ is given by
$$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18$$
\begin{enumerate}[label=(\alph*)]
\item Use the Remainder Theorem to find the remainder when $\mathrm { p } ( x )$ is divided by $x + 1$.
\item \begin{enumerate}[label=(\roman*)]
\item Use the Factor Theorem to show that $x - 3$ is a factor of $\mathrm { p } ( x )$.
\item Express $\mathrm { p } ( x )$ as a product of linear factors.
\end{enumerate}\item Sketch the curve with equation $y = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18$, stating the values of $x$ where the curve meets the $x$-axis.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2013 Q5 [10]}}