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?
+2668 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?
+289 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.
