If (a + b + c) = 0, what is the value of (a + b)^2 / ab + (b + c)^2 / bc + (c + a)^2 / ca?
Correct: C
Given a + b + c = 0.
This implies:
a + b = -c
b + c = -a
c + a = -b
Substitute these into the expression:
(a + b)^2 / ab + (b + c)^2 / bc + (c + a)^2 / ca
= (-c)^2 / ab + (-a)^2 / bc + (-b)^2 / ca
= c^2 / ab + a^2 / bc + b^2 / ca
To add these fractions, find a common denominator, which is abc.
Multiply the first term by c/c, second by a/a, third by b/b:
= (c^2 * c) / (ab * c) + (a^2 * a) / (bc * a) + (b^2 * b) / (ca * b)
= c^3 / abc + a^3 / abc + b^3 / abc
= (a^3 + b^3 + c^3) / abc
From the identity for cubes, if a + b + c = 0, then a^3 + b^3 + c^3 = 3abc.
Since a + b + c = 0 is given, we can apply this identity.
Therefore, (a^3 + b^3 + c^3) / abc = 3abc / abc = 3.