pure maths

Please refer to the following links:

http://i601.photobucket.com/albums/tt95/physicsworld9999/physicsworld01Mar160930.jpg

http://i601.photobucket.com/albums/tt95/physicsworld9999/physicsworld02Mar160930.jpg

http://i601.photobucket.com/albums/tt95/physicsworld9999/physicsworld03Mar160930.jpg


Let f(a, b, c) = a^3(b^2 - c^2) + b^3(c^2 - a^2) + c^3(a^2 - b^2)

which is a homogeneous cyclic polynomial of degree 5.

Put a = b,

f(b, b, c) = b^3(b^2 - c^2) + b^3(c^2 - b^2) + 0 = 0

Similarly, f(a, c, c) = f(a, b, a) = 0

Hence, f(a, b, c)

= (a - b)(b - c)(c - a)[k(ab + bc + ca) + l(a^2 + b^2 + c^2)]

Comparing coefficients of ab^4, l = 0

Comparing coefficients of a^2b^3, k = -1

Therefore, a^3(b^2 - c^2) + b^3(c^2 - a^2) + c^3(a^2 - b^2)

= -(a - b)(b - c)(c - a)(ab + bc + ca)