| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Given factor, find all roots |
| Difficulty | Moderate -0.5 This is a straightforward two-part question testing standard Factor and Remainder Theorem techniques. Part (a) requires simple substitution (f(3)), and part (b) involves polynomial division given one factor, then solving a quadratic—all routine C2 procedures with no problem-solving insight required. Slightly easier than average due to being purely procedural. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempting to find \(f(3)\) or \(f(-3)\) | M1 | |
| \(f(3) = 3(3)^3 - 5(3)^2 - (58\times3) + 40 = 81 - 45 - 174 + 40 = -98\) | A1 (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\{3x^3 - 5x^2 - 58x + 40 = (x-5)\}\{3x^2 + 10x - 8\}\) | M1 A1 | |
| Attempt to factorise 3-term quadratic or use quadratic formula | M1 | May be implied by correct solutions |
| \((3x-2)(x+4) = 0 \quad x = \ldots\) or \(x = \frac{-10 \pm \sqrt{100+96}}{6}\) | A1 ft | |
| \(\frac{2}{3}\) (or exact equiv.), \(-4\), \(5\) | A1 (5) | Allow 'implicit' solutions e.g. \(f(5)=0\) |
| Answer | Marks | Guidance |
|---|---|---|
| - Factorisation: \((3x^2+ax+b) = (3x+c)(x+d)\) where \( | cd | = |
# Question 2:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempting to find $f(3)$ or $f(-3)$ | M1 | |
| $f(3) = 3(3)^3 - 5(3)^2 - (58\times3) + 40 = 81 - 45 - 174 + 40 = -98$ | A1 (2) | |
**Alternative (long division):**
- Divide by $(x-3)$ to get $(3x^2 + ax + b)$, $a\neq0, b\neq0$: [M1]
- $(3x^2 + 4x - 46)$ and $-98$ seen: [A1]
- (If continues to say 'remainder = 98', isw)
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\{3x^3 - 5x^2 - 58x + 40 = (x-5)\}\{3x^2 + 10x - 8\}$ | M1 A1 | |
| Attempt to factorise 3-term quadratic or use quadratic formula | M1 | May be implied by correct solutions |
| $(3x-2)(x+4) = 0 \quad x = \ldots$ or $x = \frac{-10 \pm \sqrt{100+96}}{6}$ | A1 ft | |
| $\frac{2}{3}$ (or exact equiv.), $-4$, $5$ | A1 (5) | Allow 'implicit' solutions e.g. $f(5)=0$ |
**Notes:**
- 1st M: use of $(x-5)$ to obtain $(3x^2+ax+b)$, $a\neq0, b\neq0$
- 2nd M: attempt to factorise or solve their 3-term quadratic
- Factorisation: $(3x^2+ax+b) = (3x+c)(x+d)$ where $|cd|=|b|$
- A1ft: correct factors for their quadratic followed by a solution (at least one value, possibly incorrect), or numerically correct expression from quadratic formula
- If quadratic correctly factorised but no solutions given, last 2 marks lost
- Throughout: allow $\left(x \pm \frac{2}{3}\right)$ as alternative to $(3x \pm 2)$
**Alternative 1 (factor theorem):**
- M1: Finding $f(-4)=0$; A1: Stating $(x+4)$ is a factor
- M1: Finding third factor $(x-5)(x+4)(3x\pm2)$
- A1: Fully correct factors followed by solution; A1: All solutions correct
**Alternative 2 (direct factorisation):**
- M1: Factors $(x-5)(3x+p)(x+q)$; A1: $pq=-8$
- M1: $(x-5)(3x\pm2)(x\pm4)$; Final A marks as Alternative 12.
$$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 58 x + 40$$
\begin{enumerate}[label=(\alph*)]
\item Find the remainder when $\mathrm { f } ( x )$ is divided by ( $x - 3$ ).
Given that $( x - 5 )$ is a factor of $\mathrm { f } ( x )$,
\item find all the solutions of $\mathrm { f } ( x ) = 0$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2010 Q2 [7]}}