(a-b)^3+(b-c)^3+(c-a)^3/(a-b)(b-c)(c-a)

Jasonbradley

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Jun 16, 2024

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Factoring the Expression: (a-b)^3+(b-c)^3+(c-a)^3/(a-b)(b-c)(c-a)

This expression is a classic example of a problem that can be solved using factorization techniques. Let's break down the steps to simplify it:

Understanding the Problem

We are given the expression:

(a-b)^3 + (b-c)^3 + (c-a)^3 / (a-b)(b-c)(c-a)

Our goal is to simplify this expression by factoring it and eliminating any common terms.

Key Formula

A crucial identity to remember for this problem is the Sum of Cubes Formula:

x^3 + y^3 = (x + y)(x^2 - xy + y^2)

Applying the Formula

1. Identify the Cubes: Notice that we have three cubed terms: (a-b)^3, (b-c)^3, and (c-a)^3.

2. Apply the Formula: Let's apply the sum of cubes formula to the first two terms:

   • (a-b)^3 + (b-c)^3 = [(a-b) + (b-c)][(a-b)^2 - (a-b)(b-c) + (b-c)^2]

3. Simplify: Expanding the terms inside the brackets, we get:

   • (a-c)[a^2 - 2ab + b^2 - ab + ac + b^2 - bc + b^2 - 2bc + c^2]
   • (a-c)[a^2 + 3b^2 + c^2 - 3ab + ac - 3bc]

4. Combine with the Third Term: Now, let's add the third term, (c-a)^3:

   • (a-c)[a^2 + 3b^2 + c^2 - 3ab + ac - 3bc] + (c-a)^3

5. Factor out (c-a): We can factor out (c-a) from both terms:

   • (c-a) [(-1)(a^2 + 3b^2 + c^2 - 3ab + ac - 3bc) + (c-a)^2]

6. Simplify the Expression: Expand the remaining terms and combine like terms:

   • (c-a) [-a^2 - 3b^2 - c^2 + 3ab - ac + 3bc + c^2 - 2ac + a^2]
   • (c-a) [-3b^2 + 3ab - 3ac + 3bc]
   • (c-a) [-3(b^2 - ab + ac - bc)]
   • (c-a) [-3(b-a)(b-c)]

7. Complete the Simplification: Now, divide this expression by the denominator (a-b)(b-c)(c-a):

   • [(c-a) [-3(b-a)(b-c)]] / [(a-b)(b-c)(c-a)]
   • -3

Conclusion

Therefore, the simplified expression for (a-b)^3+(b-c)^3+(c-a)^3/(a-b)(b-c)(c-a) is -3. This simplification demonstrates the power of factorization in solving complex algebraic expressions.