| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Simple Algebraic Fraction Simplification |
| Difficulty | Moderate -0.3 Part (i) is straightforward polynomial division by a linear factor, yielding a quadratic. Part (ii) requires rearranging to use the result from (i), then solving a simple quadratic equation. This is a standard C4 exercise testing routine algebraic manipulation with clear scaffolding, 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 |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{x^3 - x^2 - 3x - 9}{x-3}\) | B1 | For factor \((x-3)\) |
| \(= \frac{(x-3)(x^2+2x+3)}{x-3}\) | B1 | |
| \(= x^2 + 2x + 3\) | B1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x^3 - x^2 - 3x - 9 = 6(x-3)\) | M1 | |
| \(\Rightarrow (x-3)(x^2+2x+3) - 6(x-3) = 0\) | A1 | |
| \(\Rightarrow (x-3)(x^2+2x-3) = 0\) | A1 | |
| \(\Rightarrow (x-3)(x+3)(x-1) = 0\) | ||
| \(\Rightarrow x = 1, 3,\) or \(-3\) | ||
| Total: 3 |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{x^3 - x^2 - 3x - 9}{x-3}$ | B1 | For factor $(x-3)$ |
| $= \frac{(x-3)(x^2+2x+3)}{x-3}$ | B1 | |
| $= x^2 + 2x + 3$ | B1 | |
| **Total: 3** | | |
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## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^3 - x^2 - 3x - 9 = 6(x-3)$ | M1 | |
| $\Rightarrow (x-3)(x^2+2x+3) - 6(x-3) = 0$ | A1 | |
| $\Rightarrow (x-3)(x^2+2x-3) = 0$ | A1 | |
| $\Rightarrow (x-3)(x+3)(x-1) = 0$ | | |
| $\Rightarrow x = 1, 3,$ or $-3$ | | |
| **Total: 3** | | |
---5 (i) Simplify $\frac { x ^ { 3 } - x ^ { 2 } - 3 x - 9 } { x - 3 }$.\\
(ii) Hence or otherwise solve the equation $x ^ { 3 } - x ^ { 2 } - 3 x - 9 = 6 ( x - 3 )$.
\hfill \mbox{\textit{OCR MEI C4 Q5 [5]}}