Standard +0.3 This is a structured Further Maths question with clear scaffolding. Writing down 8th roots of unity is direct recall (e^(2πik/8) for k=0 to 7). The verification is straightforward algebra expanding brackets. The factorization, while requiring some insight to pair conjugate roots, follows naturally from the given identity and involves standard techniques (cos(0)=1, cos(π/4)=1/√2, cos(π/2)=0). More routine than average A-level questions due to heavy scaffolding.
6 Write down all the 8th roots of unity.
Verify that
$$\left( z - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \theta } \right) \equiv z ^ { 2 } - ( 2 \cos \theta ) z + 1$$
Hence express \(z ^ { 8 } - 1\) as the product of two linear factors and three quadratic factors, where all coefficients are real and expressed in a non-trigonometric form.
6 Write down all the 8th roots of unity.
Verify that
$$\left( z - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \theta } \right) \equiv z ^ { 2 } - ( 2 \cos \theta ) z + 1$$
Hence express $z ^ { 8 } - 1$ as the product of two linear factors and three quadratic factors, where all coefficients are real and expressed in a non-trigonometric form.
\hfill \mbox{\textit{CAIE FP1 2004 Q6 [8]}}