10 The Argand diagram below shows the origin \(O\) and pentagon \(A B C D E\), where \(A , B , C , D\) and \(E\) are the points that represent the complex numbers \(a , b , c , d\) and \(e\), and where \(a\) is a positive real number. You are given that these five complex numbers are the roots of the equation \(z ^ { 5 } - a ^ { 5 } = 0\).
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- Justify each of the following statements.
(a) \(A , B , C , D\) and \(E\) lie on a circle with centre \(O\).
(b) \(A B C D E\) is a regular pentagon.
(c) \(b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c\)
(d) \(b ^ { * } = e\)
(e) \(a + b + c + d + e = 0\) - The midpoints of sides \(A B , B C , C D , D E\) and \(E A\) represent the complex numbers \(p , q , r , s\) and \(t\). Determine a polynomial equation, with real coefficients, that has roots \(p , q , r , s\) and \(t\).