| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | January |
| 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 testing routine application of factor theorem and polynomial division. Part (i) requires simple algebraic division given a known factor, part (ii) is direct substitution and expansion, and part (iii) follows immediately from understanding transformations. All steps are standard textbook exercises with no problem-solving insight required. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02w Graph transformations: simple transformations of f(x) |
## Question 13
### (i) Use the fact that $x = 2$ is a root of $f(x) = 0$ to find the exact values of the other two roots of $f(x) = 0$, expressing your answers as simply as possible. [6]
M1: Divide $f(x) = x^3 - 5x + 2$ by $(x - 2)$ using polynomial division or synthetic division
A1: Quotient is $x^2 + 2x - 1$ (or equivalent factorisation)
M1: Set $x^2 + 2x - 1 = 0$ and use quadratic formula
M1: Apply $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ correctly
A1: $x = -1 + \sqrt{2}$ and $x = -1 - \sqrt{2}$ (or equivalent simplified form)
A1: All three roots clearly identified
### (ii) Show that $f(x - 3) = x^3 - 9x^2 + 22x + 10$. [4]
M1: Substitute $x - 3$ into $f(x) = x^3 - 5x + 2$
M1: Expand $(x-3)^3$ correctly
M1: Expand and simplify $-5(x-3)$ correctly
A1: Correct final answer $f(x - 3) = x^3 - 9x^2 + 22x + 10$
### (iii) Write down the roots of $f(x - 3) = 0$. [2]
B1: If $f(x - 3) = 0$ then $f(x - 3) = 0$ means $x - 3$ equals a root of $f$
A1: Roots are $x = 5$, $x = 2 + \sqrt{2}$, $x = 2 - \sqrt{2}$ (or equivalent)13
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{7791371e-26d9-428c-8700-5121a1c6464a-4_456_387_1539_833}
\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 2007 Q13 [12]}}