| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2005 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Polynomial Division by Quadratic Divisor |
| Difficulty | Moderate -0.3 This is a straightforward polynomial division question testing standard techniques: part (i) uses the remainder theorem (simple substitution), and part (ii) requires algebraic long division of polynomials. Both are routine procedures covered early in A-level with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(x = 1\) and evaluate expression | M1 | |
| Obtain answer 8 | A1 | 2 |
| (ii) Commence division by \(x^2 + x - 1\) and obtain quotient of the form \(x + k\) | M1 | |
| Obtain quotient \(x = 1\) | A1 | |
| Obtain remainder \(2x + 4\) | A1 | |
| Correctly identify the quotient and remainder | A1/* | 4 |
(i) Substitute $x = 1$ and evaluate expression | M1 |
Obtain answer 8 | A1 | 2
(ii) Commence division by $x^2 + x - 1$ and obtain quotient of the form $x + k$ | M1 |
Obtain quotient $x = 1$ | A1 |
Obtain remainder $2x + 4$ | A1 |
Correctly identify the quotient and remainder | A1/* | 42 The polynomial $x ^ { 3 } + 2 x ^ { 2 } + 2 x + 3$ is denoted by $\mathrm { p } ( x )$.\\
(i) Find the remainder when $\mathrm { p } ( x )$ is divided by $x - 1$.\\
(ii) Find the quotient and remainder when $\mathrm { p } ( x )$ is divided by $x ^ { 2 } + x - 1$.
\hfill \mbox{\textit{CAIE P2 2005 Q2 [6]}}