| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Factorisation After Division or Remainder |
| Difficulty | Moderate -0.3 Part (i) is a straightforward polynomial division requiring algebraic manipulation to find quotient and remainder. Part (ii) requires recognizing that changing 13 to 9 means subtracting the remainder 4, allowing factorization using the quotient from part (i). This is a standard A-level technique with clear scaffolding, slightly easier than average due to the helpful 'hence' structure. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt division, or equivalent, at least as far as quotient \(2x + k\) | M1 | |
| Obtain quotient \(2x - 3\) | A1 | |
| Complete process to confirm remainder is \(4\) | A1 | [3] |
| (ii) State or imply \((4x^2 + 4x - 3)\) is a factor | B1 | |
| Obtain \((2x - 3)(2x - 1)(2x + 3)\) | B1 | [2] |
**(i)** Attempt division, or equivalent, at least as far as quotient $2x + k$ | M1 |
Obtain quotient $2x - 3$ | A1 |
Complete process to confirm remainder is $4$ | A1 | [3]
**(ii)** State or imply $(4x^2 + 4x - 3)$ is a factor | B1 |
Obtain $(2x - 3)(2x - 1)(2x + 3)$ | B1 | [2]
---3 (i) Find the quotient when the polynomial
$$8 x ^ { 3 } - 4 x ^ { 2 } - 18 x + 13$$
is divided by $4 x ^ { 2 } + 4 x - 3$, and show that the remainder is 4 .\\
(ii) Hence, or otherwise, factorise the polynomial
$$8 x ^ { 3 } - 4 x ^ { 2 } - 18 x + 9$$
\hfill \mbox{\textit{CAIE P2 2012 Q3 [5]}}