| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Known polynomial, verify then factorise |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing routine application of the Factor Theorem and polynomial division. Part (a)(i) requires simple substitution of x=-1, part (a)(ii) uses standard algebraic long division or inspection to find remaining factors, parts (b) and (c) are basic verification and sketching. All techniques are standard textbook exercises with no problem-solving insight required, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(p(-1)=(-1)^3+2(-1)^2-5(-1)-6\) | M1 | \(p(-1)\) attempted not long division |
| \(p(-1)=-1+2+5-6=0 \Rightarrow x+1\) is a factor | A1 | CSO; correctly shown \(=0\) plus statement |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Quad factor in this form: \((x^2+bx+c)\) | M1 | long division as far as constant term or comparing coefficients, or \(b=1\) or \(c=-6\) by inspection |
| \(x^2+x-6\) | A1 | correct quadratic factor |
| \([p(x)=](x+1)(x+3)(x-2)\) | A1 | must see correct product |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(p(0)=-6\); \(p(1)=-8\) | M1 | both \(p(0)\) and \(p(1)\) attempted and at least one value correct |
| \(\Rightarrow p(0)>p(1)\) | A1 | AG both values correct plus correct statement involving \(p(0)\) and \(p(1)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Cubic with one max and one min | M1 | |
| \(\bigcap\bigcup\) with \(-3,-1,2\) marked | A1 | |
| correct with minimum to right of \(y\)-axis AND going beyond \(-3\) and \(2\) | A1 |
# Question 3:
## Part (a)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $p(-1)=(-1)^3+2(-1)^2-5(-1)-6$ | M1 | $p(-1)$ attempted **not** long division |
| $p(-1)=-1+2+5-6=0 \Rightarrow x+1$ is a factor | A1 | CSO; correctly shown $=0$ plus statement |
## Part (a)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Quad factor in this form: $(x^2+bx+c)$ | M1 | long division as far as constant term **or** comparing coefficients, **or** $b=1$ **or** $c=-6$ by inspection |
| $x^2+x-6$ | A1 | correct quadratic factor |
| $[p(x)=](x+1)(x+3)(x-2)$ | A1 | **must** see correct product |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $p(0)=-6$; $p(1)=-8$ | M1 | **both** $p(0)$ and $p(1)$ attempted and at least one value correct |
| $\Rightarrow p(0)>p(1)$ | A1 | AG both values correct plus correct statement involving $p(0)$ and $p(1)$ |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Cubic with one max and one min | M1 | |
| $\bigcap\bigcup$ with $-3,-1,2$ marked | A1 | |
| correct with minimum to right of $y$-axis AND going beyond $-3$ and $2$ | A1 | |
---3 The polynomial $\mathrm { p } ( x )$ is given by
$$\mathrm { p } ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Use the Factor Theorem to show that $x + 1$ is a factor of $\mathrm { p } ( x )$.
\item Express $\mathrm { p } ( x )$ as the product of three linear factors.
\end{enumerate}\item Verify that $\mathrm { p } ( 0 ) > \mathrm { p } ( 1 )$.
\item Sketch the curve with equation $y = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6$, indicating the values where the curve crosses the $x$-axis.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2012 Q3 [10]}}