| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | One factor, one non-zero remainder |
| Difficulty | Moderate -0.3 This is a standard Factor and Remainder Theorem question requiring systematic application of two conditions to find unknowns, then factorization. It involves routine algebraic manipulation with no novel insight—slightly easier than average due to its predictable structure and clear method. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(x = -\frac{3}{2}\), equate to zero | M1 | |
| Substitute \(x = -1\) and equate to 8 | M1 | |
| Obtain a correct equation in any form | A1 | |
| Solve a relevant pair of equations for \(a\) or for \(b\) | M1 | |
| Obtain \(a = 2\) and \(b = -6\) | A1 | [5] |
| (ii) Attempt either division by \(2x + 3\) and reach a partial quotient of \(x^2 + kx\), use of an identity or observation | M1 | |
| Obtain quotient \(x^2 - 4x + 3\) | A1 | |
| Obtain linear factors \(x - 1\) and \(x - 3\) | A1 | |
| [Condone omission of repetition that \(2x + 3\) is a factor.] [If linear factors \(x - 1, x - 3\) obtained by remainder theorem or inspection, award B2 + B1.] | [3] |
(i) Substitute $x = -\frac{3}{2}$, equate to zero | M1 |
Substitute $x = -1$ and equate to 8 | M1 |
Obtain a correct equation in any form | A1 |
Solve a relevant pair of equations for $a$ or for $b$ | M1 |
Obtain $a = 2$ and $b = -6$ | A1 | [5]
(ii) Attempt either division by $2x + 3$ and reach a partial quotient of $x^2 + kx$, use of an identity or observation | M1 |
Obtain quotient $x^2 - 4x + 3$ | A1 |
Obtain linear factors $x - 1$ and $x - 3$ | A1 |
[Condone omission of repetition that $2x + 3$ is a factor.] [If linear factors $x - 1, x - 3$ obtained by remainder theorem or inspection, award B2 + B1.] | [3]4 The polynomial $a x ^ { 3 } - 5 x ^ { 2 } + b x + 9$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( 2 x + 3 )$ is a factor of $\mathrm { p } ( x )$, and that when $\mathrm { p } ( x )$ is divided by $( x + 1 )$ the remainder is 8 .\\
(i) Find the values of $a$ and $b$.\\
(ii) When $a$ and $b$ have these values, factorise $\mathrm { p } ( x )$ completely.
\hfill \mbox{\textit{CAIE P2 2013 Q4 [8]}}