Suppose that for some \(a,b,c\) we have \(a+b+c = 1\), \(ab+ac+bc = abc = -4\). What is \(a^3+b^3+c^3\)?
a + b + c = 1
a b + a c + b c = a b c
a b c = -4, solve for a, b, c
a = -2 and b = 1 and c = 2
a = -2 and b = 2 and c = 1
a = 1 and b = -2 and c = 2
a = 1 and b = 2 and c = -2
a = 2 and b = -2 and c = 1
a = 2 and b = 1 and c = -2
a^3 + b^3 + c^3 = 1 - No matter what values of a, b, c you use, you will always get 1, because +2 and -2 will always cancel out.
