Factoring the Expression (ab)³ + (bc)³ + (ca)³
This article will demonstrate how to factor the expression (ab)³ + (bc)³ + (ca)³.
Understanding the Problem
We are given a cubic expression with three terms, each involving the difference of two variables raised to the power of three. Our goal is to rewrite this expression as a product of simpler expressions, which is called factoring.
Key Identity
To factor the given expression, we'll utilize the following algebraic identity:
x³ + y³ + z³  3xyz = (x + y + z)(x² + y² + z²  xy  xz  yz)
This identity holds true for any real numbers x, y, and z.
Applying the Identity

Relating the Expression to the Identity: Notice that the given expression resembles the lefthand side of the identity. We can make it match perfectly by introducing a term
3(ab)(bc)(ca)
and then subtracting it to maintain the equivalence:(ab)³ + (bc)³ + (ca)³ = (ab)³ + (bc)³ + (ca)³  3(ab)(bc)(ca) + 3(ab)(bc)(ca)

Applying the Identity: Now, we can directly apply the identity with x = (ab), y = (bc), and z = (ca):
(ab)³ + (bc)³ + (ca)³  3(ab)(bc)(ca) + 3(ab)(bc)(ca) = [(ab) + (bc) + (ca)][(ab)² + (bc)² + (ca)²  (ab)(bc)  (ab)(ca)  (bc)(ca)] + 3(ab)(bc)(ca)

Simplifying: The first term simplifies to 0. Expanding the second term and combining like terms, we get:
3(ab)(bc)(ca) = 3(ab)(bc)(ca)
Final Result
Therefore, the factored form of the expression (ab)³ + (bc)³ + (ca)³ is:
(ab)³ + (bc)³ + (ca)³ = 3(ab)(bc)(ca)
Conclusion
We successfully factored the expression (ab)³ + (bc)³ + (ca)³ using a key algebraic identity. The resulting factored form is much simpler and easier to work with in various mathematical contexts.