| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find constants using remainder theorem |
| Difficulty | Moderate -0.3 This is a straightforward application of the Remainder Theorem to set up two simultaneous equations, followed by routine factorisation. While it requires multiple steps (substitution, solving simultaneous equations, verification, and factorisation), each individual technique is standard and the question provides clear guidance throughout. It's slightly easier than average because the method is direct with no problem-solving insight required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(x = 3\) and equate to \(30\) | M1 | |
| Substitute \(x = -1\) and equate to \(18\) | M1 | |
| Obtain a correct equation in any form | A1 | |
| Solve a relevant pair of equations for \(a\) or for \(b\) | M1 | |
| Obtain \(a = 1\) and \(b = -13\) | A1 | [5] |
| (ii) Either show that \(f(2) = 0\) or divide by \((x - 2)\), obtaining a remainder of zero | B1 | |
| Obtain quadratic factor \(2x^2 + 5x - 3\) | B1 | |
| Obtain linear factor \(2x - 1\) | B1 | |
| Obtain linear factor \(x + 3\) | B1 | |
| [Condone omission of repetition that \(x - 2\) is a factor.] | ||
| [If linear factors \(2x - 1, x + 3\) obtained by remainder theorem or inspection, award B2 + B1.] | [4] |
**(i)** Substitute $x = 3$ and equate to $30$ | M1 |
Substitute $x = -1$ and equate to $18$ | M1 |
Obtain a correct equation in any form | A1 |
Solve a relevant pair of equations for $a$ or for $b$ | M1 |
Obtain $a = 1$ and $b = -13$ | A1 | [5]
**(ii)** Either show that $f(2) = 0$ or divide by $(x - 2)$, obtaining a remainder of zero | B1 |
Obtain quadratic factor $2x^2 + 5x - 3$ | B1 |
Obtain linear factor $2x - 1$ | B1 |
Obtain linear factor $x + 3$ | B1 |
[Condone omission of repetition that $x - 2$ is a factor.] | |
[If linear factors $2x - 1, x + 3$ obtained by remainder theorem or inspection, award B2 + B1.] | [4]7 The polynomial $2 x ^ { 3 } + a x ^ { 2 } + b x + 6$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that when $\mathrm { p } ( x )$ is divided by ( $x - 3$ ) the remainder is 30 , and that when $\mathrm { p } ( x )$ is divided by ( $x + 1$ ) the remainder is 18 .\\
(i) Find the values of $a$ and $b$.\\
(ii) When $a$ and $b$ have these values, verify that ( $x - 2$ ) is a factor of $\mathrm { p } ( x )$ and hence factorise $\mathrm { p } ( x )$ completely.
\hfill \mbox{\textit{CAIE P2 2010 Q7 [9]}}