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Suppose that for some a,b,c we have a + b + c = 1, ab + ac + bc = abc = -1. What is a^3 + b^3 + c^3?

 Jul 29, 2022
 #1
avatar+2270 
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Note that \(a^3 + b^3 + c^3 = \left(\left(a+b+c\right)^2-3ab-3bc-3ac\right)\left(a+b+c\right)+3abc\) (found it off the internet)

 

However, we can rewrite this into something easier: \(\left(\left(a+b+c\right)^2-3(ab+bc+ac)\right)\left(a+b+c\right)+3abc\)

 

Can you take it from here?

 Jul 29, 2022
 #2
avatar+183 
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Builderboi has a great way to solve it but sometimes you don't have much of those formulas memorized.

Use polynomial construction.

 

a + b + c = 1

ab + ac + bc = -1

abc = -1

 

P(x) = x^3 - x^2 - x + 1

 

a, b, c are the roots of this polynomial. 

a^3 = a^2 + a - 1

b^3 = b^2 + b - 1

c^3 = c^2 + c - 1

 

a^3 + b^3 + c^3 = (a^2 + b^2 + c^2) + (a + b + c) - 3

(a + b + c)^2 - 2(ab + ac + bc) = a^2 + b^2 +  c^2 = 3

 

a^3 + b^3 + c^3 = 1.

 Jul 31, 2022

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