June 2019 Paper 2 Q17
17 Show that \(\quad (x + 1)(x + 3)(x + 4) - x(x^2 + 7x + 11)\)
can be written in the form \(\quad (x + a)(x + b) \quad\) where \(a\) and \(b\) are positive integers. [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(x^2 + 3x + x + 3\) with three terms correct or \(x^2 + 4x + k\) where \(k\) is a non-zero constant | M1 | oe expansion attempt of one pair of brackets eg1 \(\ x^2 + 4x + 3x + 12\) with three terms correct or \(x^2 + 7x + k\) where \(k\) is a non-zero constant eg2 \(\ x^2 + 4x + x + 4\) with three terms correct or \(x^2 + 5x + k\) where \(k\) is a non-zero constant |
| \(x^3 + 3x^2 + x^2 + 3x\) or \(x^3 + 4x^2 + 3x\) or \(4x^2 + 12x + 4x + 12\) or \(4x^2 + 16x + 12\) | M1dep | attempt at a full expansion with correct multiplication of their 3 or 4 terms by one of the terms in the remaining bracket oe eg \(x^3 + 4x^2 + 3x^2 + 12x\) or \(x^3 + 7x^2 + 12x\) or \(x^2 + 4x + 3x + 12\) or \(x^2 + 7x + 12\) (\(x^2 + 7x + 12\) must be from an attempt at a full expansion) or \(x^3 + 4x^2 + x^2 + 4x\) or \(x^3 + 5x^2 + 4x\) or \(3x^2 + 12x + 3x + 12\) or \(3x^2 + 15x + 12\) |
| \(x^3 + 8x^2 + 19x + 12\) | A1 | fully correct expansion allow if terms not collected eg \(x^3 + 3x^2 + x^2 + 3x + 4x^2 + 12x + 4x + 12\) or \(x^3 + 4x^2 + 3x + 4x^2 + 16x + 12\) |
| \(x^2 + 8x + 12\) | A1ft | ft M2A0 full simplification of their \((x^3 + 8x^2 + 19x + 12) - x^3 - 7x^2 - 11x\) their \((x^3 + 8x^2 + 19x + 12)\) must be a cubic |
| \(x^2 + 8x + 12\) and \((x + 6)(x + 2)\) or \((x + 2)(x + 6)\) | A1 | oe product of brackets |
Additional guidance
| 1st M1 Do not allow omissions or extras eg1 \(\ x^2 + 3x + 3\) eg2 \(\ x^2 + 3x + x + 3 + x^2\) | M0 M0 |
| For the first 2 marks terms may be seen in a grid | |
| If 1st A1 has been awarded with terms not collected, A1ft can still be awarded using their simplified cubic eg \(\ x^3 + 4x^2 + 3x + 4x^2 + 16x + 12\) \(= x^3 + 8x^2 + 18x + 12\) \(x^3 + 8x^2 + 18x + 12 - x^3 - 7x^2 - 11x\) \(= x^2 + 7x + 12\) | M1M1A1 A1ftA0 |
| First A1 may be seen embedded eg \(\ x^3 + 8x^2 + 19x + 12 - x^3 - 7x^2 - 11x\) | M1M1A1 |
| If an attempt at the expansion of all three brackets in one go is made it must be fully correct to gain M2A1, otherwise M0M0A0 eg \(\ x^2 + 3x + x + 3 + x^2 + 4x\) | M0M0A0 |
| Allow recovery of missing brackets when subtracting \(x^3 + 7x^2 + 11x\) from their cubic | |
| For final A1 allow \(x^2 + 8x + 12\) and \(a = 6 \ \ b = 2\) or \(x^2 + 8x + 12\) and \(a = 2 \ \ b = 6\) | |
| Ignore equating to zero and/or any ‘solving’ of an equation |