FP1 June 2010 Q4
4. \[\mathrm{f}(x) = x^3 + x^2 + 44x + 150\]
Given that \(\mathrm{f}(x) = (x + 3)(x^2 + ax + b)\), where \(a\) and \(b\) are real constants,
(a) find the value of \(a\) and the value of \(b\). (2)
(b) Find the three roots of \(\mathrm{f}(x) = 0\). (4)
(c) Find the sum of the three roots of \(\mathrm{f}(x) = 0\). (1)
| Scheme | Marks |
|---|---|
| \(a = -2, \quad b = 50\) | B1, B1 |
| (2) |
Notes
(a) Accept \(x^2 - 2x + 50\) as evidence of values of \(a\) and \(b\).
| Scheme | Marks |
|---|---|
| \(-3\) is a root | B1 |
| Solving 3-term quadratic \(x = \dfrac{2 \pm \sqrt{4 - 200}}{2}\) or \((x - 1)^2 - 1 + 50 = 0\) | M1 |
| \(= 1 + 7\mathrm{i}, \quad 1 - 7\mathrm{i}\) | A1, A1ft |
| (4) |
Notes
(b) B1: −3 must be seen in part (b)
M1: for solving quadratic following usual conventions
A1: for a correct root (simplified as here) and A1ft: for conjugate of first answer.
Accept correct answers with no working here.
If answers are written down as factors then isw. Must see roots for marks.
| Scheme | Marks |
|---|---|
| \((-3) + (1 + 7\mathrm{i}) + (1 - 7\mathrm{i}) = -1\) | B1ft |
| (1) | |
| 7 marks |
Notes
(c) ft requires the sum of two non-real conjugate roots and a real root resulting in a real number.
Answers including \(x\) are B0