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avatar+1827 

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.

 Dec 20, 2023
 #1
avatar+129245 
+1

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

 

And by Vieta

 

a =  (-i) (0) (i)  =  0

 

cool cool cool

 Dec 20, 2023

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