| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Finding Constants from Factor or Zero Remainder Conditions |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem and polynomial division. Part (i) requires substituting x=-2 and solving a linear equation for a. Part (ii) is routine polynomial division or synthetic division with no conceptual challenges. Both parts are standard textbook exercises requiring only direct application of well-practiced techniques. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(-2\) and equate to zero or divide by \(x + 2\) and equate remainder to zero | M1 | |
| Obtain \(a = 8\) | A1 | [2] |
| (ii) Attempt to find quotient by division or inspection or use of identity | M1 | |
| Obtain at least \(3x^2 + 2x\) | A1 | |
| Obtain \(3x^2 + 2x + 4\) with no errors seen | A1 | [3] |
**(i)** Substitute $-2$ and equate to zero or divide by $x + 2$ and equate remainder to zero | M1 |
Obtain $a = 8$ | A1 | [2]
**(ii)** Attempt to find quotient by division or inspection or use of identity | M1 |
Obtain at least $3x^2 + 2x$ | A1 |
Obtain $3x^2 + 2x + 4$ with no errors seen | A1 | [3]4 The polynomial $\mathrm { f } ( x )$ is defined by
$$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + a x + a$$
where $a$ is a constant.\\
(i) Given that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$, find the value of $a$.\\
(ii) When $a$ has the value found in part (i), find the quotient when $\mathrm { f } ( x )$ is divided by ( $x + 2$ ).
\hfill \mbox{\textit{CAIE P2 2011 Q4 [5]}}