| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Two factors given |
| Difficulty | Moderate -0.3 This is a straightforward application of the factor theorem requiring substitution of x=-1 and x=-2 to form simultaneous equations for part (i), then polynomial division for part (ii). While it involves multiple steps and techniques, each step is routine and follows standard procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks |
|---|---|
| Equate \(p(-1)\) or \(p(-2)\) to zero or divide by \((x+1)\) or \((x+2)\) and equate constant remainder to zero | M*1 |
| Obtain two equations \(a - b = 6\) and \(4a - 2b = 34\) or equivalents | A1 |
| Solve pair of equations for \(a\) or \(b\) | DM*1 |
| Obtain \(a = 11\) and \(b = 5\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply third factor is \(4x - 1\) | B1 | |
| Carry out complete expansion of \((x+1)(x+2)(4x-1)\) or \((x+1)(x+2)(Cx+D)\) | M1 | |
| Obtain \(a = 11\) | A1 | |
| Obtain \(b = 5\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Use division or equivalent and obtaining linear remainder | M1 | |
| Obtain quotient \(4x + a\), following their value of \(a\) | A1♦ | |
| Indicate remainder \(x - 13\) | A1 | [3] |
**(i) Either route:**
Equate $p(-1)$ or $p(-2)$ to zero or divide by $(x+1)$ or $(x+2)$ and equate constant remainder to zero | M*1 |
Obtain two equations $a - b = 6$ and $4a - 2b = 34$ or equivalents | A1 |
Solve pair of equations for $a$ or $b$ | DM*1 |
Obtain $a = 11$ and $b = 5$ | A1 |
**Or route:**
State or imply third factor is $4x - 1$ | B1 |
Carry out complete expansion of $(x+1)(x+2)(4x-1)$ or $(x+1)(x+2)(Cx+D)$ | M1 |
Obtain $a = 11$ | A1 |
Obtain $b = 5$ | A1 | [4]
**(ii)**
Use division or equivalent and obtaining linear remainder | M1 |
Obtain quotient $4x + a$, following their value of $a$ | A1♦ |
Indicate remainder $x - 13$ | A1 | [3]3 The polynomial $4 x ^ { 3 } + a x ^ { 2 } + b x - 2$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( x + 1 )$ and $( x + 2 )$ are factors of $\mathrm { p } ( x )$.\\
(i) Find the values of $a$ and $b$.\\
(ii) When $a$ and $b$ have these values, find the remainder when $\mathrm { p } ( x )$ is divided by $\left( x ^ { 2 } + 1 \right)$.
\hfill \mbox{\textit{CAIE P3 2014 Q3 [7]}}