| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Factorisation After Division or Remainder |
| Difficulty | Standard +0.3 This is a structured multi-part question on polynomial division with clear signposting. Part (i) is routine division/remainder theorem application, part (ii) uses the 'hence' to guide students to a straightforward factorisation, and part (iii) requires recognizing that |x| substitution gives ±√ solutions. Slightly above average due to the multi-step nature and the modulus twist, but well within standard A-level expectations with no novel problem-solving required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Carry out division, or equivalent, at least as far as \(8x^2 + kx\) | M1 | |
| Obtain correct quotient \(8x^2 + 14x - 15\) | A1 | |
| Confirm remainder is 5 | A1 | [3] |
| (ii) State or imply expression is \((x+2)\)(their quadratic quotient...) | B1√ | |
| Attempt factorisation of their quadratic quotient | M1 | |
| Obtain \((x+2)(2x+5)(4x-3)\) | A1 | [3] |
| (iii) State \(\pm\frac{3}{4}\) and no others, following their 3 linear factors | B1√ | [1] |
(i) Carry out division, or equivalent, at least as far as $8x^2 + kx$ | M1 |
Obtain correct quotient $8x^2 + 14x - 15$ | A1 |
Confirm remainder is 5 | A1 | [3]
(ii) State or imply expression is $(x+2)$(their quadratic quotient...) | B1√ |
Attempt factorisation of their quadratic quotient | M1 |
Obtain $(x+2)(2x+5)(4x-3)$ | A1 | [3]
(iii) State $\pm\frac{3}{4}$ and no others, following their 3 linear factors | B1√ | [1]4 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = 8 x ^ { 3 } + 30 x ^ { 2 } + 13 x - 25$$
(i) Find the quotient when $\mathrm { p } ( x )$ is divided by ( $x + 2$ ), and show that the remainder is 5 .\\
(ii) Hence factorise $\mathrm { p } ( x ) - 5$ completely.\\
(iii) Write down the roots of the equation $\mathrm { p } ( | x | ) - 5 = 0$.
\hfill \mbox{\textit{CAIE P2 2016 Q4 [7]}}