| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Partial Fraction Form via Division |
| Difficulty | Moderate -0.3 This is a straightforward polynomial long division question requiring students to divide a quartic by a quadratic and express the result in quotient plus remainder form. While it involves multiple steps and careful algebraic manipulation, it's a standard C3 technique with no conceptual difficulty—students simply apply the division algorithm mechanically and match coefficients. Slightly easier than average due to its routine nature. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Long division of \(2x^4 - 3x^2 + x + 1\) by \(x^2 - 1\), achieving \(2x^2\) as first term | M1 | |
| \(a = 2\) stated or implied | A1 | |
| \(c = -1\) stated or implied | A1 | |
| \(2x^2 - 1 + \frac{x}{x^2-1}\) | ||
| \(a=2, b=0, c=-1, d=1, e=0\); \(d=1\) and \(b=0, e=0\) stated or implied | A1 |
# Question 1:
| Answer/Working | Marks | Guidance |
|---|---|---|
| Long division of $2x^4 - 3x^2 + x + 1$ by $x^2 - 1$, achieving $2x^2$ as first term | M1 | |
| $a = 2$ stated or implied | A1 | |
| $c = -1$ stated or implied | A1 | |
| $2x^2 - 1 + \frac{x}{x^2-1}$ | | |
| $a=2, b=0, c=-1, d=1, e=0$; $d=1$ and $b=0, e=0$ stated or implied | A1 | |
---\begin{enumerate}
\item Given that
\end{enumerate}
$$\frac { 2 x ^ { 4 } - 3 x ^ { 2 } + x + 1 } { \left( x ^ { 2 } - 1 \right) } \equiv \left( a x ^ { 2 } + b x + c \right) + \frac { d x + e } { \left( x ^ { 2 } - 1 \right) }$$
find the values of the constants $a , b , c , d$ and $e$.\\
(4)\\
\hfill \mbox{\textit{Edexcel C3 2008 Q1 [4]}}