| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Verify factor then simplify rational expression |
| Difficulty | Moderate -0.3 This is a straightforward application of the remainder theorem and factorisation. Part (a) uses direct substitution with the remainder theorem, part (b)(i) verifies a root, part (b)(ii) requires polynomial division to find remaining factors (standard technique), and part (b)(iii) is algebraic simplification using the factorisation. All steps are routine C4 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| \((3x+2)\) is a factor | B1 | PI; use factor theorem or algebraic division to find another factor |
| Use factor theorem | M1 | |
| \(f\!\left(\frac{1}{3}\right)=0\Rightarrow(3x-1)\) is a factor | ||
| \(f(x)=(3x+2)(3x-1)(ax+b)\) | A1 | |
| \(f(x)=(3x+2)(3x-1)(3x-1)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((3x+2)\) is a factor | B1 | PI by division |
| Divide \(27x^3-9x+2\) by \((3x+2)\) | M1 | Complete division to \(ax^2+bx+c\) |
| \(9x^2-6x+1\) | A1 | |
| \(f(x)=(3x+2)(3x-1)(3x-1)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((3x+2)(3x-1)(ax+b)\) | 2 marks |
# Question 1 (Alternative Methods):
## Part 1(b)(ii) Alternative Method 1
$(3x+2)$ is a factor | B1 | PI; use factor theorem or algebraic division to find another factor
Use factor theorem | M1 |
$f\!\left(\frac{1}{3}\right)=0\Rightarrow(3x-1)$ is a factor | |
$f(x)=(3x+2)(3x-1)(ax+b)$ | A1 |
$f(x)=(3x+2)(3x-1)(3x-1)$ | A1 |
## Part 1(b)(ii) Alternative Method 2
$(3x+2)$ is a factor | B1 | PI by division
Divide $27x^3-9x+2$ by $(3x+2)$ | M1 | Complete division to $ax^2+bx+c$
$9x^2-6x+1$ | A1 |
$f(x)=(3x+2)(3x-1)(3x-1)$ | A1 |
## Part 1(b)(ii) Special Case
$(3x+2)(3x-1)(ax+b)$ | | 2 marks
---1 The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 27 x ^ { 3 } - 9 x + 2$.
\begin{enumerate}[label=(\alph*)]
\item Find the remainder when $\mathrm { f } ( x )$ is divided by $3 x + 1$.
\item \begin{enumerate}[label=(\roman*)]
\item Show that f $\left( - \frac { 2 } { 3 } \right) = 0$.
\item Express $\mathrm { f } ( x )$ as a product of three linear factors.
\item Simplify
$$\frac { 27 x ^ { 3 } - 9 x + 2 } { 9 x ^ { 2 } + 3 x - 2 }$$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2008 Q1 [9]}}