| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Combining Algebraic Fractions |
| Difficulty | Moderate -0.8 This is a straightforward algebraic fractions question requiring recognition that x²-9 = (x-3)(x+3), finding a common denominator, and simplifying. It's a routine C3 skill with clear steps and no conceptual challenges, making it easier than average but not trivial since it requires careful algebraic manipulation. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^2 - 9 = (x+3)(x-3)\) | B1 | This can occur anywhere |
| \(\frac{4x}{x^2-9} - \frac{2}{(x+3)} = \frac{4x-2(x-3)}{(x+3)(x-3)}\) | M1 | For combining two fractions with a common denominator. Denominator must be correct and at least one numerator must have been adapted. Accept as separate fractions. Condone missing brackets. Also accept \(\frac{4x(x+3)-2(x^2-9)}{(x+3)(x^2-9)}\); accept separately with condoning missing bracket; condone where only one numerator has been adapted |
| \(= \frac{2x+6}{(x+3)(x-3)}\) | A1 | A correct intermediate form of \(\frac{\text{simplified linear}}{\text{simplified quadratic}}\). Accept \(\frac{2x+6}{(x+3)(x-3)}\), \(\frac{2x+6}{x^2-9}\), and even \(\frac{(2x+6)(x+3)}{(x^2-9)(x+3)}\) |
| \(= \frac{2(x+3)}{(x+3)(x-3)}\) | ||
| \(= \frac{2}{(x-3)}\) | A1 | Further factorises and cancels (which may be implied) to reach the answer \(\frac{2}{x-3}\) |
| Total | (4) | Note: Answer of \(\frac{2}{x+3}\) probably scored from sign error giving \(\frac{2x-6}{(x+3)(x-3)}\) — would score B1 M1 A0 A0. This is not a "show that" question. |
## Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2 - 9 = (x+3)(x-3)$ | B1 | This can occur anywhere |
| $\frac{4x}{x^2-9} - \frac{2}{(x+3)} = \frac{4x-2(x-3)}{(x+3)(x-3)}$ | M1 | For combining two fractions with a common denominator. Denominator must be correct and at least one numerator must have been adapted. Accept as separate fractions. Condone missing brackets. Also accept $\frac{4x(x+3)-2(x^2-9)}{(x+3)(x^2-9)}$; accept separately with condoning missing bracket; condone where only one numerator has been adapted |
| $= \frac{2x+6}{(x+3)(x-3)}$ | A1 | A correct intermediate form of $\frac{\text{simplified linear}}{\text{simplified quadratic}}$. Accept $\frac{2x+6}{(x+3)(x-3)}$, $\frac{2x+6}{x^2-9}$, and even $\frac{(2x+6)(x+3)}{(x^2-9)(x+3)}$ |
| $= \frac{2(x+3)}{(x+3)(x-3)}$ | | |
| $= \frac{2}{(x-3)}$ | A1 | Further factorises and cancels (which may be implied) to reach the answer $\frac{2}{x-3}$ |
| **Total** | **(4)** | Note: Answer of $\frac{2}{x+3}$ probably scored from sign error giving $\frac{2x-6}{(x+3)(x-3)}$ — would score B1 M1 A0 A0. This is **not** a "show that" question. |\begin{enumerate}
\item Express $\frac { 4 x } { x ^ { 2 } - 9 } - \frac { 2 } { x + 3 }$ as a single fraction in its simplest form.\\
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2017 Q1 [4]}}