| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find constants using remainder theorem |
| Difficulty | Moderate -0.3 This is a straightforward application of the factor and remainder theorems requiring solving two simultaneous equations from f(-1)=0 and f(3)=16, followed by routine factorisation. The algebraic manipulation is simple and the method is standard textbook material, making it slightly easier than average for A-level. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(f(-1) = 0 \Rightarrow -1 - a + b = 0\) | M1, A1 | For equating their attempt at \(f(-1)\) to 0, or equiv; For the correct (unsimplified) equation |
| \(f(3) = 16 \Rightarrow 27 + 3a + b = 16\) | M1, A1, A1 | For the correct (unsimplified) equation; For equating their attempt at \(f(3)\) to 16, or equiv; For both correct values – must follow two correct equations |
| Hence \(a = -3, b = -2\) | A1 | |
| (ii) \(f(2) = 8 - 6 - 2 = 0\) | B1 | For the correct verification (from correct \(a\) & \(b\)) |
| Hence \(f(x) = (x+1)^2(x-2)\) | M1, A1 | For recognition or use of two linear factors, or full division attempt by either \((x+1)\) or \((x-2)\); For correct third factor (repeated) of \((x+1)\), and full linear factorisation stated |
**(i)** $f(-1) = 0 \Rightarrow -1 - a + b = 0$ | M1, A1 | For equating their attempt at $f(-1)$ to 0, or equiv; For the correct (unsimplified) equation
$f(3) = 16 \Rightarrow 27 + 3a + b = 16$ | M1, A1, A1 | For the correct (unsimplified) equation; For equating their attempt at $f(3)$ to 16, or equiv; For both correct values – must follow two correct equations
Hence $a = -3, b = -2$ | A1 |
**(ii)** $f(2) = 8 - 6 - 2 = 0$ | B1 | For the correct verification (from correct $a$ & $b$)
Hence $f(x) = (x+1)^2(x-2)$ | M1, A1 | For recognition or use of two linear factors, or full division attempt by either $(x+1)$ or $(x-2)$; For correct third factor (repeated) of $(x+1)$, and full linear factorisation stated
---5 The cubic polynomial $\mathrm { f } ( x )$ is given by
$$f ( x ) = x ^ { 3 } + a x + b$$
where $a$ and $b$ are constants. It is given that ( $x + 1$ ) is a factor of $\mathrm { f } ( x )$ and that the remainder when $\mathrm { f } ( x )$ is divided by $( x - 3 )$ is 16 .\\
(i) Find the values of $a$ and $b$.\\
(ii) Hence verify that $\mathrm { f } ( 2 ) = 0$, and factorise $\mathrm { f } ( x )$ completely.
\hfill \mbox{\textit{OCR C2 2005 Q5 [8]}}