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Let a, b, c be complex numbers, with a real, such that

 

a+b+c = ab + ac + bc = abc = 3.

 

Find a.

 

I see that b = d + ei and c = d - ei, for some real d and e.

I used vietas and got a system of three equations in a, d and e, which idk how to solve :(

 Jul 19, 2022

Best Answer 

 #1
avatar+197 
+1

Use polynomial construction!
S_1 = 3

S_2 = 3

S_3 = 3


We can then construct a polyoimial where a, b, c are roots of this polynomial.

 

P(x) = x^3 - 3x^2 + 3x - 3

= (x - 1)^3 - 2

 

(x-1)^3 = 2

Since I believe a has to be real based on the fact that b = d + ei and c = d - ei , \(a = \sqrt[3]{2} + 1\)

 Jul 19, 2022
 #1
avatar+197 
+1
Best Answer

Use polynomial construction!
S_1 = 3

S_2 = 3

S_3 = 3


We can then construct a polyoimial where a, b, c are roots of this polynomial.

 

P(x) = x^3 - 3x^2 + 3x - 3

= (x - 1)^3 - 2

 

(x-1)^3 = 2

Since I believe a has to be real based on the fact that b = d + ei and c = d - ei , \(a = \sqrt[3]{2} + 1\)

Voldemort Jul 19, 2022
 #2
avatar+37 
+1

OHHHHHHH thank you! How come i didn't see!!!

SupersonicMan12  Jul 19, 2022

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