| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Single polynomial, two remainder/factor conditions |
| Difficulty | Moderate -0.3 This is a standard C2 Factor/Remainder Theorem question requiring systematic application of f(3)=55 and f(-1)=-9 to form simultaneous equations, then factorising using the given factor. While it involves multiple steps (6 marks typical), each step follows routine procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempting \(f(\pm1)\) or \(f(\pm3)\) | M1 | Numbers substituted into expression |
| Sets \(f(3)=55\), i.e. \(27a-99+3b+4=55\) | A1 | Applying \(f(3)\) correctly and setting equal to 55 |
| Sets \(f(-1)=-9\), i.e. \(-a-11-b+4=-9\) | A1 | Applying \(f(-1)\) correctly and setting equal to \(-9\) |
| Solve simultaneously for \(a\) or \(b\) | M1 | May make slip |
| \(a=6\) and \(b=-4\) | A1 cao | Both values correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(f(x)=(3x+2)(2x^2-5x+2)\) or \((x+\frac{2}{3})(6x^2-15x+6)\) | M1 A1 | Attempting to divide cubic by \((3x+2)\) or \((x+\frac{2}{3})\) leading to 3TQ beginning with correct term |
| \(=(3x+2)(x-2)(2x-1)\) or \(=(3x+2)(2-x)(1-2x)\) | M1 A1 | 2nd M1: valid attempt to factorise quadratic; 2nd A1: all three factors correct together |
## Question 4:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempting $f(\pm1)$ or $f(\pm3)$ | M1 | Numbers substituted into expression |
| Sets $f(3)=55$, i.e. $27a-99+3b+4=55$ | A1 | Applying $f(3)$ correctly and setting equal to 55 |
| Sets $f(-1)=-9$, i.e. $-a-11-b+4=-9$ | A1 | Applying $f(-1)$ correctly and setting equal to $-9$ |
| Solve simultaneously for $a$ or $b$ | M1 | May make slip |
| $a=6$ and $b=-4$ | A1 cao | Both values correct |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $f(x)=(3x+2)(2x^2-5x+2)$ or $(x+\frac{2}{3})(6x^2-15x+6)$ | M1 A1 | Attempting to divide cubic by $(3x+2)$ or $(x+\frac{2}{3})$ leading to 3TQ beginning with correct term |
| $=(3x+2)(x-2)(2x-1)$ or $=(3x+2)(2-x)(1-2x)$ | M1 A1 | 2nd M1: valid attempt to factorise quadratic; 2nd A1: all three factors correct together |
---4. $\mathrm { f } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } + b x + 4$, where $a$ and $b$ are constants.
When $\mathrm { f } ( x )$ is divided by ( $x - 3$ ) the remainder is 55\\
When $\mathrm { f } ( x )$ is divided by $( x + 1 )$ the remainder is - 9
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$ and the value of $b$.
Given that $( 3 x + 2 )$ is a factor of $\mathrm { f } ( x )$,
\item factorise $\mathrm { f } ( x )$ completely.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2013 Q4 [9]}}