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Find the sum of all complex values of $a,$ such that the polynomial $x^4 + (a^2 - 1) x^2 + a^3$ has exactly two distinct complex roots.

 
 Jan 9, 2021
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The polynomial must be of the form (x - r)^2*(x - s)^2.  Expanding and comparing coefficients:

-2r - 2s = 0

r^2 + 4rs + s^2 = a^2 - 1

-2rs^2 - 2r^2s = 0

r^2s^2 = a^3

 

Solving this system, the sum of all possible values of a is 7.

 
 Jan 9, 2021

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