Find all real numbers a such that the roots of the polynomial x^3 + x + a
form an arithmetic progression and are not all real.
x^3 +0x^2 + x + a
Let the roots be
m - d , m , m + d
By Vieta
The sum of the roots = 0
So
3m = 0
m = 0
And by Vieta
(m - d)m + (m - d)(m + d) + (m + d) m = 1
-d^2 = 1
d^2 = -1
d = i , d = -i
So.....the roots are
-i , 0 , i
a = (-i) (0) (i) = 0
+1911 
