| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Unknown constant, verify 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 or inspection to complete the factorization. Both parts are routine textbook exercises with clear methods and minimal problem-solving demand, 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 |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute \(x = -2\) into expression and equate to zero; Obtain \(-32 + 4a + 2(a+1) - 18 = 0\) or equivalent; Obtain \(a = 8\) | M1, A1, A1 | [3] |
| Attempt to find quadratic factor by division, inspection, etc.; Obtain \(4x^2 - 9\); State \((x+2)(2x-3)(2x+3)\) | M1, A1, A1 | [3] |
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $x = -2$ into expression and equate to zero; Obtain $-32 + 4a + 2(a+1) - 18 = 0$ or equivalent; Obtain $a = 8$ | M1, A1, A1 | [3] |
| Attempt to find quadratic factor by division, inspection, etc.; Obtain $4x^2 - 9$; State $(x+2)(2x-3)(2x+3)$ | M1, A1, A1 | [3] |
---\begin{enumerate}[label=(\roman*)]
\item Given that $(x + 2)$ is a factor of
$$4x^3 + ax^2 - (a + 1)x - 18,$$
find the value of the constant $a$. [3]
\item When $a$ has this value, factorise $4x^3 + ax^2 - (a + 1)x - 18$ completely. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2015 Q2 [6]}}