| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Polynomial Division by Quadratic Divisor |
| Difficulty | Moderate -0.8 This question tests routine polynomial division and direct application of the factor theorem. Part (i) is a standard algebraic long division exercise requiring careful arithmetic but no problem-solving insight. Part (ii) is straightforward substitution to verify f(-1)=0. Both parts are mechanical procedures commonly practiced in textbooks, making this easier than average for A-level. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Commence division by \(x^2 + x - 1\) obtaining quotient of the form \(x + k\) | M1 | |
| Obtain quotient \(x + 2\) | A1 | |
| Obtain remainder \(3x + 4\) | A1 | |
| Identify the quotient and remainder correctly | A1√ | [4] |
| (ii) Substitute \(x = -1\) and evaluate expression | M1 | |
| Obtain answer \(0\) | A1 | [2] |
**(i)** Commence division by $x^2 + x - 1$ obtaining quotient of the form $x + k$ | M1 |
Obtain quotient $x + 2$ | A1 |
Obtain remainder $3x + 4$ | A1 |
Identify the quotient and remainder correctly | A1√ | [4]
**(ii)** Substitute $x = -1$ and evaluate expression | M1 |
Obtain answer $0$ | A1 | [2]4 The polynomial $x ^ { 3 } + 3 x ^ { 2 } + 4 x + 2$ is denoted by $\mathrm { f } ( x )$.\\
(i) Find the quotient and remainder when $\mathrm { f } ( x )$ is divided by $x ^ { 2 } + x - 1$.\\
(ii) Use the factor theorem to show that $( x + 1 )$ is a factor of $\mathrm { f } ( x )$.
\hfill \mbox{\textit{CAIE P2 2010 Q4 [6]}}