| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2003 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Known polynomial, verify then factorise |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem requiring routine algebraic manipulation. Part (i) involves substituting x=-1 and solving for a (standard 2-mark work), while part (ii) requires verification by substitution then polynomial division and factorising a quadratic—all standard techniques with no problem-solving insight needed. Easier than average A-level questions. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks |
|---|---|
| EITHER: Substitute \(-1\) for \(x\) and equate to zero | M1 |
| Obtain answer \(a = 6\) | A1 |
| OR: Carry out complete division and equate remainder to zero | M1 |
| Obtain answer \(a = 6\) | A1 |
| Answer | Marks |
|---|---|
| Substitute \(6\) for \(a\) and either show \(f(x) = 0\) or divide by \((x-2)\) obtaining a remainder of zero | B1 |
| State or imply \((x+1)(x-2) = x^2 - x - 2\) | B1 |
| Attempt to find another quadratic factor by division or inspection | M1 |
| State factor \((x^2 + x - 3)\) | A1 |
| OR: Obtain \(x^3 + 2x^2 - 2x - 3\) after division by \(x + 1\), or \(x^3 - x^2 - 5x + 6\) after division by \(x - 2\) | B1 |
| Attempt to find a quadratic factor by further division by relevant divisor or by inspection | M1 |
| State factor \((x^2 + x - 3)\) | A1 |
### (i)
**EITHER:** Substitute $-1$ for $x$ and equate to zero | M1 |
Obtain answer $a = 6$ | A1 |
**OR:** Carry out complete division and equate remainder to zero | M1 |
Obtain answer $a = 6$ | A1 |
**Total: [2]**
### (ii)
Substitute $6$ for $a$ and either show $f(x) = 0$ or divide by $(x-2)$ obtaining a remainder of zero | B1 |
State or imply $(x+1)(x-2) = x^2 - x - 2$ | B1 |
Attempt to find another quadratic factor by division or inspection | M1 |
State factor $(x^2 + x - 3)$ | A1 |
**OR:** Obtain $x^3 + 2x^2 - 2x - 3$ after division by $x + 1$, or $x^3 - x^2 - 5x + 6$ after division by $x - 2$ | B1 |
Attempt to find a quadratic factor by further division by relevant divisor or by inspection | M1 |
State factor $(x^2 + x - 3)$ | A1 |
**Total: [4]**The polynomial $x^4 - 6x^2 + x + a$ is denoted by $f(x)$.
\begin{enumerate}[label=(\roman*)]
\item It is given that $(x + 1)$ is a factor of $f(x)$. Find the value of $a$. [2]
\item When $a$ has this value, verify that $(x - 2)$ is also a factor of $f(x)$ and hence factorise $f(x)$ completely. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2003 Q3 [6]}}