| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2002 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find constant then factorise |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem requiring substitution of x=-2 to find 'a', followed by polynomial division and factorising a quadratic. Both parts are routine textbook exercises with no problem-solving insight needed, making it easier than average but not trivial due to the algebraic manipulation required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER: Substitute \(-2\) for \(x\) and equate to zero | M1 | |
| Obtain answer \(a = 7\) | A1 | |
| OR: Carry out complete division and equate remainder to zero | M1 | |
| Obtain answer \(a = 7\) | A1 | 2 |
| (ii) EITHER: Find quadratic factor by division or inspection | M1 | |
| Obtain answer \(3x^2 + x - 4\) | A1 | |
| Factorise completely to \((x + 2)(x - 1)(3x + 4)\) | A1 | |
| [To earn the M1 the quotient (or factor) must contain \(3x^2\) and another term, at least.] | ||
| OR: State \((x - 1)\) is a factor | B1 | |
| Find remaining linear factor by division or by inspection | M1 | |
| Factorise completely to \((x + 2)(x - 1)(3x + 4)\) | A1 | 3 |
**(i) EITHER:** Substitute $-2$ for $x$ and equate to zero | M1 |
Obtain answer $a = 7$ | A1 |
**OR:** Carry out complete division and equate remainder to zero | M1 |
Obtain answer $a = 7$ | A1 | 2
**(ii) EITHER:** Find quadratic factor by division or inspection | M1 |
Obtain answer $3x^2 + x - 4$ | A1 |
Factorise completely to $(x + 2)(x - 1)(3x + 4)$ | A1 |
[To earn the M1 the quotient (or factor) must contain $3x^2$ and another term, at least.] | |
**OR:** State $(x - 1)$ is a factor | B1 |
Find remaining linear factor by division or by inspection | M1 |
Factorise completely to $(x + 2)(x - 1)(3x + 4)$ | A1 | 3
---2 The cubic polynomial $3 x ^ { 3 } + a x ^ { 2 } - 2 x - 8$ is denoted by $\mathrm { f } ( x )$.\\
(i) Given that ( $x + 2$ ) is a factor of $\mathrm { f } ( x )$, find the value of $a$.\\
(ii) When $a$ has this value, factorise $\mathrm { f } ( x )$ completely.
\hfill \mbox{\textit{CAIE P2 2002 Q2 [5]}}