| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Single unknown constant |
| Difficulty | Easy -1.2 This is a straightforward application of the factor theorem requiring substitution of x=-2 to find a constant, followed by polynomial division to find the quadratic factor. Both parts are routine textbook exercises with no problem-solving insight needed, making it easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(x = -2\) and equate to zero, or divide by \(x + 2\) and equate constant remainder to zero, or use a factor \(Ax^2 + Bx + C\) and reach an equation in \(a\) | M1 | |
| Obtain answer \(a = 4\) | A1 | Total: 2 marks |
| (ii) Attempt to find quadratic factor by division or inspection | M1 | |
| State or exhibit quadratic factor \(x^2 - 2x + 2\) | A1 | Total: 2 marks |
(i) Substitute $x = -2$ and equate to zero, or divide by $x + 2$ and equate constant remainder to zero, or use a factor $Ax^2 + Bx + C$ and reach an equation in $a$ | M1 |
Obtain answer $a = 4$ | A1 | **Total: 2 marks**
(ii) Attempt to find quadratic factor by division or inspection | M1 |
State or exhibit quadratic factor $x^2 - 2x + 2$ | A1 | **Total: 2 marks**
**Guidance notes:**
- [The M1 is earned if division reaches a partial quotient $x^2 + kx$, or if inspection has an unknown factor $x^2 + bx + c$ and an equation in $b$ and/or $c$, or if inspection without working states two coefficients with the correct moduli.]
---2 The polynomial $x ^ { 3 } - 2 x + a$, where $a$ is a constant, is denoted by $\mathrm { p } ( x )$. It is given that ( $x + 2$ ) is a factor of $\mathrm { p } ( x )$.\\
(i) Find the value of $a$.\\
(ii) When $a$ has this value, find the quadratic factor of $\mathrm { p } ( x )$.
\hfill \mbox{\textit{CAIE P3 2007 Q2 [4]}}