Higher June 2025 Paper 2 Q13
13 Show that \((2x + 3)(x - 1)(x + 2)\) can be written in the form \(ax^3 + bx^2 + cx + d\) where \(a\), \(b\), \(c\) and \(d\) are integers. (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| Shown | M1 | for a method to find the product of any two linear expressions (3 out of 4 terms correct or 4 terms ignoring signs) eg \(2x^2 - 2x + 3x - 3\ (= 2x^2 + x - 3)\) or \(x^2 - x + 2x - 2\ (= x^2 + x - 2)\) or \(2x^2 + 4x + 3x + 6\ (= 2x^2 + 7x + 6)\) |
| M1 | (dep on M1) for a complete method to obtain all terms, half of which are correct (ft their first product) eg \(2x^3 - 2x^2 + 3x^2 + 4x^2 - 3x - 4x + 6x - 6\) or \(2x^3 - 2x^2 + 4x^2 - 4x + 3x^2 - 3x + 6x - 6\) or \(2x^3 + 4x^2 + 3x^2 + 6x - 2x^2 - 4x - 3x - 6\) or \(2x^3 + 4x^2 + x^2 - 3x + 2x - 6\) or \(2x^3 + 2x^2 + 3x^2 - 4x + 3x - 6\) or \(2x^3 - 2x^2 + 7x^2 - 7x + 6x - 6\) | |
| C1 | for \(2x^3 + 5x^2 - x - 6\) from correct working |
Additional guidance
Note that, for example, \(2x^2 + x\) in the expansion of \((2x + 3)(x - 1)\) is regarded as 3 correct terms
First product must be quadratic with at least 3 terms but need not be simplified or may be simplified incorrectly
Accept \(a = 2, b = 5, c = -1, d = -6\)
Condone \(-1x\)