| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | One factor, one non-zero remainder |
| Difficulty | Moderate -0.8 This is a straightforward application of the Factor and Remainder Theorems requiring students to set up two simultaneous equations by substituting x = -1/2 (factor) and x = -2 (remainder), then solve for a and b. Part (ii) involves routine factorisation once constants are known. Standard textbook exercise with clear method and minimal problem-solving demand. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(x = -\frac{1}{2}\), equate to zero and obtain a correct equation, e.g. \(-\frac{1}{4} + \frac{5}{4} - \frac{1}{2}a + b = 0\) | B1 | |
| Substitute \(x = -2\) and equate to \(9\) | M1 | |
| Obtain a correct equation, e.g. \(-16 + 20 - 2a + b = 9\) | A1 | |
| Solve for \(a\) or for \(b\) | M1 | |
| Obtain \(a = -4\) and \(b = -3\) | A1 | [5] |
| (ii) Attempt division by \(2x + 1\) reaching a partial quotient of \(x^2 + kx\) | M1 | |
| Obtain quadratic factor \(x^2 + 2x - 3\) | A1 | |
| Obtain factorisation \((2x + 1)(x + 3)(x - 1)\) | A1 | [3] |
**(i)** Substitute $x = -\frac{1}{2}$, equate to zero and obtain a correct equation, e.g. $-\frac{1}{4} + \frac{5}{4} - \frac{1}{2}a + b = 0$ | B1 |
Substitute $x = -2$ and equate to $9$ | M1 |
Obtain a correct equation, e.g. $-16 + 20 - 2a + b = 9$ | A1 |
Solve for $a$ or for $b$ | M1 |
Obtain $a = -4$ and $b = -3$ | A1 | [5]
**(ii)** Attempt division by $2x + 1$ reaching a partial quotient of $x^2 + kx$ | M1 |
Obtain quadratic factor $x^2 + 2x - 3$ | A1 |
Obtain factorisation $(2x + 1)(x + 3)(x - 1)$ | A1 | [3]
[The M1 is earned if inspection has an unknown factor of $x^2 + cx + f$ and an equation in $c$ and/or $f$, or if two coefficients with the correct moduli are stated without working.] [If linear factors are found by the factor theorem, give B1 + B1 for $(x - 1)$ and $(x + 3)$, and then B1 for the complete factorisation.]5 The polynomial $2 x ^ { 3 } + 5 x ^ { 2 } + a x + b$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( 2 x + 1 )$ is a factor of $\mathrm { p } ( x )$ and that when $\mathrm { p } ( x )$ is divided by $( x + 2 )$ the remainder is 9 .\\
(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 P3 2010 Q5 [8]}}