| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find quotient and remainder by division |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing routine algebraic manipulation and polynomial division. Part (i) is simple substitution, part (ii) is binomial expansion and simplification, and part (iii) is standard polynomial division or synthetic division—all mechanical procedures with no novel insight required. Slightly easier than average due to the guided structure. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \(p(1) = 1^4 - (1-2)^4 = 1 - 1 = 0 \therefore (x-1)\) is a factor | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \(p(x) = x^4 - [x^4 + 4x^3(-2) + 6x^2(-2)^2 + 4x(-2)^3 + (-2)^4]\) | M1 A1 | |
| \(= x^4 - [x^4 - 8x^3 + 24x^2 - 32x + 16]\) | M1 | |
| \(= 8x^3 - 24x^2 + 32x - 16\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| Long division of \(8x^3 - 24x^2 + 32x - 16\) by \((x+1)\) giving \(8x^2 - 32x + 64\) remainder \(-80\) | M2 | |
| quotient \(= 8x^2 - 32x + 64\) | A1 | |
| remainder \(= -80\) | A1 | (10) |
# Question 8:
## Part (i):
| Answer/Working | Marks | Notes |
|---|---|---|
| $p(1) = 1^4 - (1-2)^4 = 1 - 1 = 0 \therefore (x-1)$ is a factor | M1 A1 | |
## Part (ii):
| Answer/Working | Marks | Notes |
|---|---|---|
| $p(x) = x^4 - [x^4 + 4x^3(-2) + 6x^2(-2)^2 + 4x(-2)^3 + (-2)^4]$ | M1 A1 | |
| $= x^4 - [x^4 - 8x^3 + 24x^2 - 32x + 16]$ | M1 | |
| $= 8x^3 - 24x^2 + 32x - 16$ | A1 | |
## Part (iii):
| Answer/Working | Marks | Notes |
|---|---|---|
| Long division of $8x^3 - 24x^2 + 32x - 16$ by $(x+1)$ giving $8x^2 - 32x + 64$ remainder $-80$ | M2 | |
| quotient $= 8x^2 - 32x + 64$ | A1 | |
| remainder $= -80$ | A1 | **(10)** |
---8. $\mathrm { p } ( x ) = x ^ { 4 } - ( x - 2 ) ^ { 4 }$.\\
(i) Show that ( $x - 1$ ) is a factor of $\mathrm { p } ( x )$.\\
(ii) Show that
$$\mathrm { p } ( x ) = 8 x ^ { 3 } - 24 x ^ { 2 } + 32 x - 16$$
(iii) Find the quotient and remainder when $\mathrm { p } ( x )$ is divided by ( $x + 1$ ).\\
\hfill \mbox{\textit{OCR C2 Q8 [10]}}