Understanding the (ab)^3+(bc)^3+(ca)^3 Formula
The formula (ab)^3+(bc)^3+(ca)^3 = 3(ab)(bc)(ca) is a useful algebraic identity that helps simplify expressions and solve problems. Let's explore this formula and its applications in detail.
Understanding the Formula
This formula states that the sum of the cubes of the differences between three variables (a, b, and c) is equal to three times the product of the differences between each pair of variables.
Derivation of the Formula
The formula can be derived using the following steps:

Factorization: We can factorize the expression using the algebraic identity for the sum of cubes:
 x^3 + y^3 = (x+y)(x^2  xy + y^2)

Applying the identity:
 (ab)^3 + (bc)^3 + (ca)^3 = [(ab)+(bc)][(ab)^2  (ab)(bc) + (bc)^2] + (ca)^3

Simplifying:
 = (ac)[(ab)^2  (ab)(bc) + (bc)^2] + (ca)^3
 = (ac)[(a^2  2ab + b^2)  (ab  ac  b^2 + bc) + (b^2  2bc + c^2)] + (ca)^3
 = (ac)[a^2  3ab + 3b^2  3bc + c^2] + (ca)^3
 = (ac)[(a^2  3ab + 3b^2  3bc + c^2) + (c^2  2ac + a^2)]
 = (ac)[2a^2  3ab + 3b^2  3bc  2ac + 2c^2]
 = (ac)[(ac)(2a + 2c)  3b(ac)]
 = 3(ab)(bc)(ca)
Applications of the Formula
This formula finds its application in various mathematical fields:
 Simplifying Algebraic Expressions: The formula allows us to simplify complex expressions involving the cubes of differences.
 Solving Equations: It can be utilized to solve equations where the unknown variable is involved in a cubic expression.
 Polynomial Factorization: The formula provides a way to factorize polynomials of degree three.
Example
Let's consider an example to see the application: Simplify the expression: (xy)^3 + (yz)^3 + (zx)^3
Using the formula, we have: (xy)^3 + (yz)^3 + (zx)^3 = 3(xy)(yz)(zx)
This simplifies the expression considerably.
Conclusion
The formula (ab)^3+(bc)^3+(ca)^3 = 3(ab)(bc)(ca) is a valuable tool in simplifying algebraic expressions, solving equations, and factoring polynomials. Its understanding and application can significantly enhance problemsolving skills in various mathematical contexts.