| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2020 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Fully specified polynomial: verify factor and solve |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem requiring students to verify a given factor by substitution, then perform polynomial division and solve a quadratic. All steps are routine with no problem-solving insight needed, making it easier than average but not trivial since it requires multiple standard techniques. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| DR \(f(-2) = (-2)^3 + (-2)^2 - 8(-2) - 12 = 0\) | M1 | Substitution seen. Do not allow division here |
| so [by the factor theorem] \((x+2)\) is a factor | A1 [2] | Clear conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| DR \(f(x) = (x+2)(x^2 - x - 6)\) | M1, A1 | M1: Attempt to divide or factorise. A1: Correct quadratic factor seen. Also allow M1 A1 for \(f(x) = (x-3)(x^2+4x+4)\) if \((x-3)\) also established as a factor by division or factor theorem |
| \(f(x) = (x+2)^2(x-3) = 0\) | B1 | Product of linear factors seen |
| so \(x = 3\) or \(x = -2\) [repeated] | A1 [4] | Do not allow without full working |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| **DR** $f(-2) = (-2)^3 + (-2)^2 - 8(-2) - 12 = 0$ | M1 | Substitution seen. Do not allow division here |
| so [by the factor theorem] $(x+2)$ is a factor | A1 [2] | Clear conclusion |
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## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| **DR** $f(x) = (x+2)(x^2 - x - 6)$ | M1, A1 | M1: Attempt to divide or factorise. A1: Correct quadratic factor seen. Also allow M1 A1 for $f(x) = (x-3)(x^2+4x+4)$ if $(x-3)$ also established as a factor by division or factor theorem |
| $f(x) = (x+2)^2(x-3) = 0$ | B1 | Product of linear factors seen |
| so $x = 3$ or $x = -2$ [repeated] | A1 [4] | Do not allow without full working |
---7 In this question you must show detailed reasoning.
The function $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12$ for all values of $x$.
\begin{enumerate}[label=(\alph*)]
\item Use the factor theorem to show that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$.
\item Solve the equation $\mathrm { f } ( x ) = 0$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 1 2020 Q7 [6]}}