+132 Let a, b, c be the roots of the cubic polynomial x^3 - x - 1 = 0. Find a(b - c)^2 + b(c - a)^2 + c(a - b)^2.
thank you!
+129852 We can write the polynomial as
x^3 + 0x^2 - 1x - 1
Sum of the roots = 0
So a + b + c = 0 implies that
a + b = - c
a + c - b
b + c = -a
Product of the roots = 1
So
abc = 1
a(b - c)2^2 = ab^2 - 2abc + ac^2
b(c - a)^2 = bc^2 - 2abc + ba^2
c(a - b)^2 = ca^2 - 2abc + cb^2
Sum of this can be written as = ab ( a + b) + ac ( a + c) + bc ( b + c) - 6abc
Substituting we have
ab (-c) + ac ( - b) + bc ( -a) - 6abc =
- abc - abc - abc - 6abc =
-9 (abc) =
-9 (1) =
-9
