| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find remainder(s) then factorise |
| Difficulty | Moderate -0.3 This is a standard C2 polynomial question testing routine techniques: remainder theorem (direct substitution), factor theorem verification, and factorization to solve a cubic. Part (iii) requires factoring out (x-3) and solving the resulting quadratic, likely using the quadratic formula. While it's a multi-part question worth 9 marks total, all steps are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(-8 - 36 - 14 + 33 = -25\) | M1, A1 (2 marks) | Substitute \(x = -2\), or attempt complete division by \((x + 2)\); Obtain \(-25\), as final answer |
| (ii) \(27 - 81 + 21 + 33 = 0\) A.G. | B1 (1 mark) | Confirm \(f(3) = 0\), or equiv using division |
| (iii) \(x = 3\) \(f(x) = (x-3)(x^2 - 6x - 11)\) | B1, M1, A1, A1 (4 marks) | State \(x = 3\) as a root at any point; Attempt complete division by \((x-3)\) or equiv; Obtain \(x^2 - 6x + k\); Obtain completely correct quotient |
| \(x = \frac{6 ± \sqrt{36 + 44}}{2} = 3 ± 2\sqrt{5}\) or \(3 ± \sqrt{20}\) | M1, A1 (6 marks) | Attempt use of quadratic formula, or equiv, to find roots; Obtain \(3 ± 2\sqrt{5}\) or \(3 ± \sqrt{20}\) |
**(i)** $-8 - 36 - 14 + 33 = -25$ | M1, A1 (2 marks) | Substitute $x = -2$, or attempt complete division by $(x + 2)$; Obtain $-25$, as final answer
**(ii)** $27 - 81 + 21 + 33 = 0$ A.G. | B1 (1 mark) | Confirm $f(3) = 0$, or equiv using division
**(iii)** $x = 3$ $f(x) = (x-3)(x^2 - 6x - 11)$ | B1, M1, A1, A1 (4 marks) | State $x = 3$ as a root at any point; Attempt complete division by $(x-3)$ or equiv; Obtain $x^2 - 6x + k$; Obtain completely correct quotient
$x = \frac{6 ± \sqrt{36 + 44}}{2} = 3 ± 2\sqrt{5}$ or $3 ± \sqrt{20}$ | M1, A1 (6 marks) | Attempt use of quadratic formula, or equiv, to find roots; Obtain $3 ± 2\sqrt{5}$ or $3 ± \sqrt{20}$
**Total: 9 marks**The polynomial f(x) is defined by $f(x) = x^3 - 9x^2 + 7x + 33$.
\begin{enumerate}[label=(\roman*)]
\item Find the remainder when f(x) is divided by $(x + 2)$. [2]
\item Show that $(x - 3)$ is a factor of f(x). [1]
\item Solve the equation f(x) = 0, giving each root in an exact form as simply as possible. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2007 Q8 [9]}}