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ehrenfest
#1
May17-08, 10:54 AM
P: 1,996
1. The problem statement, all variables and given/known data
Find the zeros of the polynomial

[tex]P(x) = x^4-6x^3+18x^2-30x+25[/tex]

knowing that the sum of two of them is 4.


2. Relevant equations
http://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formulas


3. The attempt at a solution
Let x_1,x_2,x_3,x_4 be the complex roots and let x_1 +x_2 = 4. Here are the Viete relations in this case:

[tex]x_1+x_2+x_3+x_4 = 6 [/tex]

[tex] x_1 x_2 +x_1 x_3 +x_1 x_4 + x_2 x_3 + x_2 x_4 +x_3 x_4 = 18 [/tex]

[tex] x_1 x_2 x_3 + x_1 x_3 x_4 +x_2 x_3 x_4 +x_1 x_2 x_4= 30 [/tex]

[tex] x_1 x_2 x_3 x_4 = 25[/tex]

The first one implies that x_3 +x_4 =2. And then the second one implies that x_1 x_2 + x_3 x_4 = 10 but that is as far as I can get.

Please just give a hint.
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