| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | One factor, one non-zero remainder |
| Difficulty | Moderate -0.8 This is a straightforward application of the Factor and Remainder Theorems requiring students to set up two simple equations (p(-1)=0 and p(2)=12) to find a and b, then factorize. It's routine bookwork with clear signposting and standard algebraic manipulation, making it easier than average but not trivial since it requires multiple steps and careful arithmetic. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks |
|---|---|
| Substitute \(x = -1\) and equate to zero obtaining e.g. \((-1)^3 - (-1)^2 + a(-1) + b = 0\) | B1 |
| Substitute \(x = 2\) and equate to \(12\) | M1 |
| Obtain a correct 3-term equation | A1 |
| Solve a relevant pair of equations for \(a\) or \(b\) | M1 |
| Obtain \(a = 2\) and \(b = 4\) | A1 |
| Answer | Marks |
|---|---|
| Attempt division by \(x + 1\) reaching a partial quotient of \(x^2 + kx\), or similar stage by inspection | M1 |
| Obtain quadratic factor \(x^2 - 2x - 4\) | A1 |
**(i)**
| Substitute $x = -1$ and equate to zero obtaining e.g. $(-1)^3 - (-1)^2 + a(-1) + b = 0$ | B1 |
| Substitute $x = 2$ and equate to $12$ | M1 |
| Obtain a correct 3-term equation | A1 |
| Solve a relevant pair of equations for $a$ or $b$ | M1 |
| Obtain $a = 2$ and $b = 4$ | A1 |
**Total: 5 marks**
**(ii)**
| Attempt division by $x + 1$ reaching a partial quotient of $x^2 + kx$, or similar stage by inspection | M1 |
| Obtain quadratic factor $x^2 - 2x - 4$ | A1 |
(Ignore failure to repeat that $x + 1$ is a factor)
**Total: 2 marks**
---4 The polynomial $x ^ { 3 } - x ^ { 2 } + a x + b$ is denoted by $\mathrm { p } ( x )$. It is given that ( $x + 1$ ) is a factor of $\mathrm { p } ( x )$ and that when $\mathrm { p } ( x )$ is divided by $( x - 2 )$ the remainder is 12 .\\
(i) Find the values of $a$ and $b$.\\
(ii) When $a$ and $b$ have these values, factorise $\mathrm { p } ( x )$.
\hfill \mbox{\textit{CAIE P2 2005 Q4 [7]}}