| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Combining Algebraic Fractions |
| Difficulty | Moderate -0.3 Part (a) requires factorising both numerator and denominator then cancelling common factors - a standard algebraic skill. Part (b) involves combining fractions with different denominators and simplifying, which is routine but requires careful algebraic manipulation. This is a typical C3 algebraic fractions question with no novel insight required, making it slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{(3x+2)(x-1)}{(x+1)(x-1)} = \frac{3x+2}{x+1}\) | M1B1A1 (3) | M1: attempt to factorise numerator; B1: factorising denominator seen anywhere; A1: given answer. If factorisation of denom. not seen, correct answer implies B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{3x+2}{x+1} - \frac{1}{x(x+1)} = \frac{x(3x+2)-1}{x(x+1)}\) | M1 | Expressing over common denominator |
| \(3x^2 + 2x - 1 \equiv (3x-1)(x+1)\) | M1 | Multiplying out numerator and attempt to factorise |
| Answer: \(\frac{3x-1}{x}\) | A1 (3) |
# Question 1:
## Part (a)
$\frac{(3x+2)(x-1)}{(x+1)(x-1)} = \frac{3x+2}{x+1}$ | M1B1A1 (3) | M1: attempt to factorise numerator; B1: factorising denominator seen anywhere; A1: given answer. If factorisation of denom. not seen, correct answer implies B1
## Part (b)
$\frac{3x+2}{x+1} - \frac{1}{x(x+1)} = \frac{x(3x+2)-1}{x(x+1)}$ | M1 | Expressing over common denominator
$3x^2 + 2x - 1 \equiv (3x-1)(x+1)$ | M1 | Multiplying out numerator and attempt to factorise
Answer: $\frac{3x-1}{x}$ | A1 (3) |
---\begin{enumerate}[label=(\alph*)]
\item Simplify $\frac { 3 x ^ { 2 } - x - 2 } { x ^ { 2 } - 1 }$.
\item Hence, or otherwise, express $\frac { 3 x ^ { 2 } - x - 2 } { x ^ { 2 } - 1 } - \frac { 1 } { x ( x + 1 ) }$ as a single fraction in its simplest form.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2006 Q1 [6]}}