October 2020 Paper 1 Q7
7 In this question you must show detailed reasoning.
The function \(\mathrm{f}(x)\) is defined by \(\mathrm{f}(x) = x^3 + x^2 - 8x - 12\) for all values of \(x\).
(a) Use the factor theorem to show that \((x+2)\) is a factor of \(\mathrm{f}(x)\). [2]
(b) Solve the equation \(\mathrm{f}(x) = 0\). [4]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\mathrm{f}(-2) = (-2)^3 + (-2)^2 - 8(-2) - 12 = 0\) | M1 | 1.1a |
| so [by the factor theorem] \((x+2)\) is a factor | A1 | 2.2a |
| [2] |
Notes
M1: Substitution seen. Do not allow for division here
A1: Clear conclusion.
| Scheme | Marks | AO |
|---|---|---|
| DR \(\mathrm{f}(x) = (x+2)\left(x^2 - x - 6\right)\) | M1 A1 | 3.1a 1.1 |
| \(\mathrm{f}(x) = (x+2)^2(x-3) = 0\) | B1 | 1.1 |
| so \(x = 3\) or \(x = -2\) [repeated] | A1 | 2.1 |
| [4] |
Notes
M1: Attempt to divide or factorise
A1: Correct quadratic factor seen
B1: Product of linear factors seen
A1: Do not allow without full working
Also allow M1 A1 for \(\mathrm{f}(x) = (x-3)\left(x^2+4x+4\right)\) if \((x-3)\) also established as a factor by division or factor theorem.