FP1 June 2009 Q3
3. \[\mathrm{f}(x) = (x^2 + 4)(x^2 + 8x + 25)\]
(a) Find the four roots of \(\mathrm{f}(x) = 0\). (5)
(b) Find the sum of these four roots. (2)
| Scheme | Marks |
|---|---|
| \(x^2 + 4 = 0 \quad \Rightarrow \quad x = k\mathrm{i}, \quad x = \pm 2\mathrm{i}\) | M1, A1 |
| Solving 3-term quadratic | M1 |
| \(x = \dfrac{-8 \pm \sqrt{64 - 100}}{2} = -4 + 3\mathrm{i}\) and \(-4 - 3\mathrm{i}\) | A1 A1ft |
| (5) |
Notes
(a) Just \(x = 2\mathrm{i}\) is M1 A0
\(x = \pm 2\) is M0A0
M1 for solving quadratic follows usual conventions, then A1 for a correct root (simplified as here) and A1ft for conjugate of first answer.
Accept correct answers with no working here. Do not give accuracy marks for factors unless followed by roots.
| Scheme | Marks |
|---|---|
| \(2\mathrm{i} + (-2\mathrm{i}) + (-4 + 3\mathrm{i}) + (-4 - 3\mathrm{i}) = -8\) | M1 A1cso |
| (2) | |
| [7] |
Alternative method
| Scheme | Marks |
|---|---|
| Expands \(\mathrm{f}(x)\) as quartic and chooses \(\pm\) coefficient of \(x^3\) | M1 |
| \(-8\) | A1 cso |
Notes
(b) M1 for adding four roots of which at least two are complex conjugates and getting a real answer. A1 for −8 following correct roots or the alternative method. If any incorrect working in part (a) this A mark will be A0