PURE MATH Q

(a) As the degree of f(x) is odd, it has at least one real root.(b) Since |α_i| = 1, the real root in (a) should be -1 or 1 By Vieta's formula, α_1 α_2 α_3 α_4 α_5 = 1

If they are both real and equal to -1, then α_1 α_2 α_3 α_4 α_5 = -1, a contradiction.If two are complex (α_1 α_2) , the other three are real and equal to -1, then α_1 α_2 is positive and α_3 α_4 α_5 = -1, a contradiction.

If four are complex (α_1 α_2 α_3 α_4) , the other one are real and equal to -1, then α_1 α_2 α_3 α_4 is positive and α_5 = -1, a contradiction.

So,1 must be a root of f(x).

f(1) = a_0 + a_1 + a_2 + a_3 + a_4 + a_5 = 0