| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | June |
| 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 substitution of x=-2 to verify the factor, followed by polynomial division and factorising a quadratic. It's routine C2 content with clear steps and no problem-solving insight needed, making it easier than average but not trivial since it requires accurate algebraic manipulation across multiple steps. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(-2) = 2\cdot(-2)^3 - 7\cdot(-2)^2 - 10\cdot(-2) + 24\) | M1 | Attempts \(f(\pm 2)\). Long division is M0 |
| \(= 0\) so \((x+2)\) is a factor | A1 | For \(=0\) and conclusion. Stating "hence factor" or "it is a factor" or a tick or "QED" is fine |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x) = (x+2)(2x^2 - 11x + 12)\) | M1 A1 | 1st M1: Attempts long division by correct factor or other method to obtain \((2x^2 \pm ax \pm b)\), \(a \neq 0\), \(b \neq 0\) |
| \(f(x) = (x+2)(2x-3)(x-4)\) | dM1 A1 | 2nd M1: Factorise quadratic — dependent on previous M1. A1 needs all three factors together. Note: \(f(x)=(x+2)(x-1.5)(x-4)\) loses last A1 |
# Question 4:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(-2) = 2\cdot(-2)^3 - 7\cdot(-2)^2 - 10\cdot(-2) + 24$ | M1 | Attempts $f(\pm 2)$. Long division is M0 |
| $= 0$ so $(x+2)$ is a factor | A1 | For $=0$ **and conclusion**. Stating "hence factor" or "it is a factor" or a tick or "QED" is fine |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) = (x+2)(2x^2 - 11x + 12)$ | M1 A1 | 1st M1: Attempts long division by correct factor or other method to obtain $(2x^2 \pm ax \pm b)$, $a \neq 0$, $b \neq 0$ |
| $f(x) = (x+2)(2x-3)(x-4)$ | dM1 A1 | 2nd M1: Factorise quadratic — dependent on previous M1. A1 needs all three factors together. Note: $f(x)=(x+2)(x-1.5)(x-4)$ loses last A1 |
---4.
$$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 10 x + 24$$
\begin{enumerate}[label=(\alph*)]
\item Use the factor theorem to show that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$.
\item Factorise f(x) completely.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2012 Q4 [6]}}