| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | June |
| 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 straightforward C1 question testing standard application of Factor/Remainder Theorem with routine algebraic manipulation. All parts follow predictable patterns: direct substitution for remainder, verification of a given factor, polynomial division/factorisation, and basic curve sketching. No problem-solving insight required, making it easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Remainder \(= p(1) = 1 + 1 - 8 - 12 = -18\) | M1, A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(p(-2) = -8 + 4 + 16 - 12 = 0 \Rightarrow (x + 2)\) is factor | M1, A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Quad factor by comparing coefficients or \((x^2 + kx + 6)\) by inspection | M1 | |
| \(p(x) = (x + 2)(x^2 - x - 6)\) | A1 | |
| \(p(x) = (x + 2)^2(x - 3)\) or \((x+2)(x+2)(x-3)\) | A1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \((k =) -12\) | B1 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Cubic shape (one max and one min) Maximum at \((-2, 0)\) and through \((3, 0)\) – at least one of these values marked | M1, A1, A1 | 3 |
**6(a)**
| Remainder $= p(1) = 1 + 1 - 8 - 12 = -18$ | M1, A1 | 2 | Use of $p(1)$ NOT long division |
**6(b)(i)**
| $p(-2) = -8 + 4 + 16 - 12 = 0 \Rightarrow (x + 2)$ is factor | M1, A1 | 2 | NOT long division $p(-2)$ shown $= 0$ and statement |
**6(b)(ii)**
| Quad factor by comparing coefficients or $(x^2 + kx + 6)$ by inspection | M1 | | Or full long division or attempt at Factor Theorem using $f(\pm 3)$ |
| $p(x) = (x + 2)(x^2 - x - 6)$ | A1 | |
| $p(x) = (x + 2)^2(x - 3)$ or $(x+2)(x+2)(x-3)$ | A1 | 3 | CSO; SC: B1 for $(x+2)(x***))(x-3)$ by inspection or without working |
**6(c)(i)**
| $(k =) -12$ | B1 | 1 | Condone $y = -12$ or $(0, -12)$ |
**6(c)(ii)**
| Cubic shape (one max and one min) Maximum at $(-2, 0)$ and through $(3, 0)$ – at least one of these values marked | M1, A1, A1 | 3 | "correct" graph as shown (touching smoothly at $-2, 3$ marked and minimum to right of $y$-axis) |
---6 The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12$.
\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 $x + 2$ is a factor of $\mathrm { p } ( x )$.
\item Express $\mathrm { p } ( x )$ as the product of linear factors.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item The curve with equation $y = x ^ { 3 } + x ^ { 2 } - 8 x - 12$ passes through the point $( 0 , k )$. State the value of $k$.
\item Sketch the graph of $y = x ^ { 3 } + x ^ { 2 } - 8 x - 12$, indicating the values of $x$ where the curve touches or crosses the $x$-axis.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2008 Q6 [11]}}