| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Verify factor then factorise/solve |
| Difficulty | Easy -1.2 This is a straightforward application of the remainder theorem and polynomial division with no problem-solving required. Part (a) uses direct substitution of x=1/4, part (b)(i) applies the factor theorem identically, and part (b)(ii) is routine algebraic long division or coefficient comparison. All techniques are standard C4 content with no conceptual challenges. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Use \(x = \frac{1}{8}\) in evaluation: \(f(\frac{1}{8}) = 8 \times \frac{1}{8} + 6 \times \frac{1}{16} - 14 \times \frac{1}{4} - 1 = -4\) | M1 A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(g(x) = \text{number}(s) + d = 0\); \(d = 3\) | M1 A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(g(x) = (4x-1)(2x^2 + bx - 3)\); \(a = 2\), \(c = -3\); F on \(d\) (where \(c = -d\)) | B1F M1 A1 | 3 marks |
**1(a)**
Use $x = \frac{1}{8}$ in evaluation: $f(\frac{1}{8}) = 8 \times \frac{1}{8} + 6 \times \frac{1}{16} - 14 \times \frac{1}{4} - 1 = -4$ | M1 A1 | 2 marks | NMS 2/2; no ISW
**1(b)(i)**
$g(x) = \text{number}(s) + d = 0$; $d = 3$ | M1 A1 | 2 marks | Use factor theorem to find $d$; See some processing; NMS 2/2
**1(b)(ii)**
$g(x) = (4x-1)(2x^2 + bx - 3)$; $a = 2$, $c = -3$; F on $d$ (where $c = -d$) | B1F M1 A1 | 3 marks | Any appropriate method; PI; NMS 2/2
---\begin{enumerate}[label=(\alph*)]
\item The polynomial $f(x)$ is defined by $f(x) = 8x^3 + 6x^2 - 14x - 1$.
Find the remainder when $f(x)$ is divided by $(4x - 1)$. [2 marks]
\item The polynomial $g(x)$ is defined by $g(x) = 8x^3 + 6x^2 - 14x + d$.
\begin{enumerate}[label=(\roman*)]
\item Given that $(4x - 1)$ is a factor of $g(x)$, find the value of the constant $d$. [2 marks]
\item Given that $g(x) = (4x - 1)(ax^2 + bx + c)$, find the values of the integers $a$, $b$ and $c$. [3 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2010 Q1 [7]}}