| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Single unknown from factor condition |
| Difficulty | Moderate -0.8 Part (i) is a direct application of the factor theorem requiring substitution of x=-2 and solving a linear equation for a. Part (ii) involves factorizing the cubic and analyzing a quadratic discriminant. This is a standard textbook exercise with routine techniques and no novel problem-solving required, making it easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(-2\) and equate to zero or divide by \(x + 2\) and equate remainder to zero or use \(-2\) in synthetic division | M1 | |
| Obtain \(a = -1\) | A1 | [2] |
| (ii) Attempt to find quadratic factor by division reaching \(x^2 + kx\), or inspection as far as \((x+2)(x^2 + Bx + c)\) and equations for one or both of \(B\) and \(C\), or \((x+2)(Ax^2 + Bx + 7)\) and equations for one or both of \(A\) and \(B\) | M1 | |
| Obtain \(x^2 - 3x + 7\) | A1 | |
| Use discriminant to obtain \(-19\), or equivalent, and confirm one root | A1 cwo | [3] |
**(i)** Substitute $-2$ and equate to zero or divide by $x + 2$ and equate remainder to zero or use $-2$ in synthetic division | M1 |
Obtain $a = -1$ | A1 | [2]
**(ii)** Attempt to find quadratic factor by division reaching $x^2 + kx$, or inspection as far as $(x+2)(x^2 + Bx + c)$ and equations for one or both of $B$ and $C$, or $(x+2)(Ax^2 + Bx + 7)$ and equations for one or both of $A$ and $B$ | M1 |
Obtain $x^2 - 3x + 7$ | A1 |
Use discriminant to obtain $-19$, or equivalent, and **confirm one root** | A1 cwo | [3]3 The polynomial $\mathrm { f } ( x )$ is defined by
$$f ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 14$$
where $a$ is a constant. It is given that ( $x + 2$ ) is a factor of $\mathrm { f } ( x )$.\\
(i) Find the value of $a$.\\
(ii) Show that, when $a$ has this value, the equation $\mathrm { f } ( x ) = 0$ has only one real root.
\hfill \mbox{\textit{CAIE P3 2013 Q3 [5]}}