Standard +0.3 This is a structured multi-part question on roots of unity that follows standard Further Maths patterns. While it requires knowledge of De Moivre's theorem and complex conjugate pairs, each part guides students through familiar techniques: listing roots of unity, multiplying conjugate pairs to get real quadratics, and factorizing polynomials. The final part requires recognizing that x^6 - x^3 + 1 relates to 9th roots of unity, which is slightly less routine but still a standard FP1 exercise. Overall, this is slightly easier than average for A-level due to its structured, guided nature.
State the fifth roots of unity in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(- \pi < \theta \leqslant \pi\).
Simplify
$$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right)$$
Hence find the real factors of
$$x ^ { 5 } - 1$$
Express the six roots of the equation
$$x ^ { 6 } - x ^ { 3 } + 1 = 0$$
as three conjugate pairs, in the form \(\cos \theta \pm \mathrm { i } \sin \theta\).
Hence find the real factors of
$$x ^ { 6 } - x ^ { 3 } + 1$$