| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2009 |
| Session | November |
| Marks | 6 |
| 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 = -1/2 to find 'a', followed by polynomial division or inspection to complete the factorisation. Both parts are routine procedures with no conceptual challenges beyond standard AS-level technique. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(x = -\frac{1}{2}\) and equate to zero | M1 | |
| Obtain \(a = -11\) | A1 | [2 marks] |
| (ii) EITHER: Attempt division by \(2x + 1\) reaching a partial quotient \(2x^2 - 5x\) | M1 | |
| Obtain quadratic factor \(2x^2 - 5x - 3\) | A1 | |
| Obtain complete factorisation \((2x + 1)^2(x - 3)\) | A1 + A1 | |
| OR: Obtain factor \((x - 3)\) by inspection or factor theorem | B2 | |
| Attempt division by \((x - 3)\) reaching a partial quotient \(4x^2 + 4x\) | M1 | |
| Obtain complete factorisation \((2x + 1)^2(x - 3)\) | A1 | [4 marks] |
**(i)** Substitute $x = -\frac{1}{2}$ and equate to zero | M1 |
Obtain $a = -11$ | A1 | [2 marks]
**(ii)** **EITHER:** Attempt division by $2x + 1$ reaching a partial quotient $2x^2 - 5x$ | M1 |
Obtain quadratic factor $2x^2 - 5x - 3$ | A1 |
Obtain complete factorisation $(2x + 1)^2(x - 3)$ | A1 + A1 |
**OR:** Obtain factor $(x - 3)$ by inspection or factor theorem | B2 |
Attempt division by $(x - 3)$ reaching a partial quotient $4x^2 + 4x$ | M1 |
Obtain complete factorisation $(2x + 1)^2(x - 3)$ | A1 | [4 marks]3 The polynomial $4 x ^ { 3 } - 8 x ^ { 2 } + a x - 3$, where $a$ is a constant, is denoted by $\mathrm { p } ( x )$. It is given that ( $2 x + 1$ ) is a factor of $\mathrm { p } ( x )$.\\
(i) Find the value of $a$.\\
(ii) When $a$ has this value, factorise $\mathrm { p } ( x )$ completely.
\hfill \mbox{\textit{CAIE P2 2009 Q3 [6]}}