Challenging +1.2 This is a multi-part Further Maths question requiring knowledge of roots of unity, substitution to transform equations, and working with complex numbers in polar form. While the first part is routine (5th roots of unity), the substitution w-1=z and subsequent analysis requires problem-solving beyond standard exercises. The final part about arguments of roots with smaller modulus adds geometric insight. Moderately challenging for Further Maths but follows established patterns.
9 Write down, in any form, all the roots of the equation \(z ^ { 5 } - 1 = 0\).
Hence find all the roots of the equation
$$( w - 1 ) ^ { 4 } + ( w - 1 ) ^ { 3 } + ( w - 1 ) ^ { 2 } + w = 0$$
and deduce that none of them is real.
Find the arguments of the two roots which have the smaller modulus.
9 Write down, in any form, all the roots of the equation $z ^ { 5 } - 1 = 0$.
Hence find all the roots of the equation
$$( w - 1 ) ^ { 4 } + ( w - 1 ) ^ { 3 } + ( w - 1 ) ^ { 2 } + w = 0$$
and deduce that none of them is real.
Find the arguments of the two roots which have the smaller modulus.
\hfill \mbox{\textit{CAIE FP1 2007 Q9 [10]}}