| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Given factor, find all roots |
| Difficulty | Moderate -0.8 This is a straightforward C1 question on factor theorem and polynomial manipulation. Part (i) involves routine polynomial division given a known factor, then solving a quadratic. Parts (ii) and (iii) are direct algebraic substitution and translation of roots. All techniques are standard with no problem-solving insight required, making it easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks |
|---|---|
| M1 | \(x - 2\) is factor so |
| M1 | attempt at division by \(x - 2\) as far as \(x^3 - 2x^2\) seen in working |
| A1 | \(x^2 + 2x - 1\) obtained |
| M1 | attempt at quadratic formula or completing the square |
| A2 | \(-1 \pm \sqrt{2}\) as final answer |
| Answer | Marks |
|---|---|
| B1 | \(f(x - 3) = (x - 3)^3 - 5(x - 3) + 2\) |
| M1 | \((x - 3)(x^2 - 6x + 9)\) or other constructive attempt at expanding |
| A1 | \((x - 3)^3\) eg \(1 \, 3 \, 3 \, 1\) soi |
| B1 | \(x^3 - 9x^2 + 27x - 27\) |
| B1 | \(-5x + 15\) \([+2]\) |
# Question 2
## (i)
M1 | $x - 2$ is factor so
M1 | attempt at division by $x - 2$ as far as $x^3 - 2x^2$ seen in working
A1 | $x^2 + 2x - 1$ obtained
M1 | attempt at quadratic formula or completing the square
A2 | $-1 \pm \sqrt{2}$ as final answer
**Guidance:** eg may be implied by division or other factor $(x^2 \ldots -1)$ or $(x^2 + 2x \ldots)$ or B3 www; ft their quadratic; $-2 \pm \sqrt{8}$ or $2 \pm 2\sqrt{2}$; A1 for seen; or B3 www
## (ii)
B1 | $f(x - 3) = (x - 3)^3 - 5(x - 3) + 2$
M1 | $(x - 3)(x^2 - 6x + 9)$ or other constructive attempt at expanding
A1 | $(x - 3)^3$ eg $1 \, 3 \, 3 \, 1$ soi
B1 | $x^3 - 9x^2 + 27x - 27$
B1 | $-5x + 15$ $[+2]$
**Guidance:** for attempt at multiplying out 2 brackets or valid attempt at multiplying all 3; alt: A2 for correct full unsimplified expansion or A1 for correct 2 bracket expansion eg $(x - 5)(x^2 - 4x + 2)$; condone factors here, not roots; if B0 in this part, allow SC1 for their roots in (i) $-3$2
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{618d118a-2557-42f3-9b55-4a55dda93a97-1_449_376_631_889}
\captionsetup{labelformat=empty}
\caption{Fig. 13}
\end{center}
\end{figure}
Fig. 13 shows a sketch of the curve $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = x ^ { 3 } - 5 x + 2$.\\
(i) Use the fact that $x = 2$ is a root of $\mathrm { f } ( x ) = 0$ to find the exact values of the other two roots of $\mathrm { f } ( x ) = 0$, expressing your answers as simply as possible.\\
(ii) Show that $\mathrm { f } ( x - 3 ) = x ^ { 3 } - 9 x ^ { 2 } + 22 x - 10$.\\
(iii) Write down the roots of $\mathrm { f } ( x - 3 ) = 0$.
\hfill \mbox{\textit{OCR MEI C1 Q2 [12]}}