June 2023 Paper 3 Q4
4 In this question you must show detailed reasoning.
Find the coordinates of the points where the curve \(y = x^3 - 2x^2 - 5x + 6\) crosses the \(x\)-axis. [4]
| Scheme | Marks | AO |
|---|---|---|
| DR \(x^3 - 2x^2 - 5x + 6 = 0\) For \(x = 1\), \(1^3 - 2(1)^2 - 5(1) + 6 = 0\) oe so \(x = 1\) is a root or \((x-1)\) is a factor | M1 | 1.1a |
| \((x-1)\left(x^2 - x - 6\right)\) | M1 | 1.1 |
| \((x-1)(x-3)(x+2)\) | DM1 | 1.1 |
| (1, 0), (3, 0), (–2, 0) | A1 | 1.1 |
| [4] |
Notes
M1: Finding one root or factor by factor theorem or division to a remainder of 0. A conclusion is needed for this M1.
M1: Factorising to find quadratic factor or division (method seen) or factor theorem again (substitution shown) to get a different root/factor
DM1: Completion, all 3 factors/roots seen, dep on previous M1
Might not be in same place
A1: All 3 points as coordinates or pairs of values
Dep on M3