| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2009 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Partial Fraction Form via Division |
| Difficulty | Moderate -0.8 This is a straightforward application of the Remainder Theorem followed by routine polynomial long division. Part (a) requires simple substitution of x=1/3, and part (b) is a standard algebraic division exercise with no conceptual challenges—both are textbook procedures requiring only careful arithmetic. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(f\left(\frac{1}{3}\right) = 3 \times \frac{1}{27} + 8 \times \frac{1}{9} - 3 \times \frac{1}{3} - 5\) | M1 | Use \(\frac{1}{3}\) in evaluating \(f(x)\) |
| \(= -5\) | A1 | No ISW; Evidence of Remainder Theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Long division giving \(x^2\) and an \(x\) term; \(x^2 + px\) seen | M1 | Division with \(x^2\) and an \(x\) term seen |
| \(a=1\), \(b=3\) or \(x^2 + 3x + \frac{c}{3x-1}\) | A1 | Explicit or in expression |
| \(c = -5\) | B1 | Condone \(+\frac{-5}{3x-1}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\frac{(3x-1)(x^2+px)}{3x-1} - \frac{5}{3x-1}\) | (M1) | Split fraction and attempt factors |
| \(x^2 + 3x\) | (A1) | \(a=1\), \(b=3\) |
| \(-\frac{5}{3x-1}\) | (B1) | \(c=-5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(f(x) = 3ax^3 + (3b-a)x^2 - bx + c\) | (M1) | Multiply by \((3x-1)\) and attempt to collect terms |
| \(a=1\), \(b=3\) | (A1) | |
| \(c=-5\) | (B1) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(f(x) = (ax^2+bx)(3x-1)+c\) | (M1) | Multiply by \((3x-1)\) and attempt to find \(a\), \(b\), \(c\) |
| \(x=0 \Rightarrow c=-5\) | (B1) | Substitute 3 values of \(x\); form 3 simultaneous equations |
| \(x=1 \Rightarrow 2a+2b+c=3\); \(x=2 \Rightarrow 20a+10b+c=45\) | ||
| \(a=1\), \(b=3\) | (A1) |
# Question 1:
## Part (a)
| Working | Marks | Guidance |
|---------|-------|----------|
| $f\left(\frac{1}{3}\right) = 3 \times \frac{1}{27} + 8 \times \frac{1}{9} - 3 \times \frac{1}{3} - 5$ | M1 | Use $\frac{1}{3}$ in evaluating $f(x)$ |
| $= -5$ | A1 | No ISW; Evidence of Remainder Theorem |
**Total: 2 marks**
## Part (b)
| Working | Marks | Guidance |
|---------|-------|----------|
| Long division giving $x^2$ and an $x$ term; $x^2 + px$ seen | M1 | Division with $x^2$ and an $x$ term seen |
| $a=1$, $b=3$ or $x^2 + 3x + \frac{c}{3x-1}$ | A1 | Explicit or in expression |
| $c = -5$ | B1 | Condone $+\frac{-5}{3x-1}$ |
**Alternative 1:**
| Working | Marks | Guidance |
|---------|-------|----------|
| $\frac{(3x-1)(x^2+px)}{3x-1} - \frac{5}{3x-1}$ | (M1) | Split fraction and attempt factors |
| $x^2 + 3x$ | (A1) | $a=1$, $b=3$ |
| $-\frac{5}{3x-1}$ | (B1) | $c=-5$ |
**Alternative 2:**
| Working | Marks | Guidance |
|---------|-------|----------|
| $f(x) = 3ax^3 + (3b-a)x^2 - bx + c$ | (M1) | Multiply by $(3x-1)$ and attempt to collect terms |
| $a=1$, $b=3$ | (A1) | |
| $c=-5$ | (B1) | |
**Alternative 3:**
| Working | Marks | Guidance |
|---------|-------|----------|
| $f(x) = (ax^2+bx)(3x-1)+c$ | (M1) | Multiply by $(3x-1)$ and attempt to find $a$, $b$, $c$ |
| $x=0 \Rightarrow c=-5$ | (B1) | Substitute 3 values of $x$; form 3 simultaneous equations |
| $x=1 \Rightarrow 2a+2b+c=3$; $x=2 \Rightarrow 20a+10b+c=45$ | | |
| $a=1$, $b=3$ | (A1) | |
**Total: 5 marks**
---1
\begin{enumerate}[label=(\alph*)]
\item Use the Remainder Theorem to find the remainder when $3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5$ is divided by $3 x - 1$.
\item Express $\frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5 } { 3 x - 1 }$ in the form $a x ^ { 2 } + b x + \frac { c } { 3 x - 1 }$, where $a , b$ and $c$ are integers.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2009 Q1 [5]}}