| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find constant then factorise |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem requiring substitution to find 'a', then polynomial division or comparison of coefficients to factorise completely. It's more routine than average A-level questions since the method is directly signposted and involves standard algebraic manipulation with no problem-solving insight required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| DR \(5 \times 2^3 - 4 \times 2^2 + 2a - 2 = 0\) oe, or \(40 - 16 + 2a - 2 = 0\) oe | M1 | Substitute \(x=2\) and equate to 0. May be implied, or \(\div\) by \((x-2)\) & obtain \(5x^2+6x+1\) |
| \(a = -11\) | A1 | \(a=-11\), with no working SC: B1 |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| DR \(5x^3 - 4x^2 - 11x - 2 = (x-2)(px^2 + qx + r)\) oe | M1 | attempted, or attempt \((5x^3 - 4x^2 - 11x - 2) \div (x-2)\) ft (a). May be implied by next line |
| \(= (x-2)(5x^2 + 6x + 1)\) | A1 | |
| \(= (x-2)(5x+1)(x+1)\) ISW | A1 | Above method must be seen. Use of solutions: M0A0A0 |
| [3] |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| **DR** $5 \times 2^3 - 4 \times 2^2 + 2a - 2 = 0$ oe, or $40 - 16 + 2a - 2 = 0$ oe | M1 | Substitute $x=2$ and equate to 0. May be implied, or $\div$ by $(x-2)$ & obtain $5x^2+6x+1$ |
| $a = -11$ | A1 | $a=-11$, with no working SC: B1 |
| **[2]** | | |
---
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| **DR** $5x^3 - 4x^2 - 11x - 2 = (x-2)(px^2 + qx + r)$ oe | M1 | attempted, or attempt $(5x^3 - 4x^2 - 11x - 2) \div (x-2)$ ft (a). May be implied by next line |
| $= (x-2)(5x^2 + 6x + 1)$ | A1 | |
| $= (x-2)(5x+1)(x+1)$ ISW | A1 | Above method must be seen. Use of solutions: M0A0A0 |
| **[3]** | | |
---2 In this question you must show detailed reasoning.\\
The cubic polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 5 x ^ { 3 } - 4 x ^ { 2 } + a x - 2$, where $a$ is a constant.
You are given that $( x - 2 )$ is a factor of $\mathrm { f } ( x )$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$.
\item Find all the factors of $\mathrm { f } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q2 [5]}}