(a-b)^3 + (b-c)^3 + (c-a)^3/9(a-b)(b-c)(c-a)

3 min read Jun 16, 2024

Factoring and Simplifying the Expression (a-b)^3 + (b-c)^3 + (c-a)^3 / 9(a-b)(b-c)(c-a)

This expression involves the sum of cubes, which can be factored using a specific formula. Let's break down the steps to simplify it.

1. Factor the Numerator

The numerator of this expression follows the pattern of the sum of cubes factorization:

x³ + y³ = (x + y)(x² - xy + y²)

We can apply this to each term in the numerator:

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

Simplifying further:

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

Now, the numerator becomes:

(a - c)[(a - b)² - (a - b)(b - c) + (b - c)²] + 2(c - a)[(c - a)²]

2. Combine Terms and Simplify

We can factor out (c - a) from both terms in the numerator:

(c - a) [(a - b)² - (a - b)(b - c) + (b - c)² + 2(c - a)²]

Expanding the squared terms and simplifying:

(c - a) [(a² - 2ab + b²) - (ab - ac - b² + bc) + (b² - 2bc + c²) + 2(c² - 2ac + a²)]

(c - a) [3a² + 3b² + 3c² - 3ab - 3ac - 3bc]

Finally, we can factor out 3:

(c - a) * 3(a² + b² + c² - ab - ac - bc)

3. Simplify the Entire Expression

Now, the expression becomes:

(c - a) * 3(a² + b² + c² - ab - ac - bc) / 9(a - b)(b - c)(c - a)

We can cancel out (c - a) and simplify further:

3(a² + b² + c² - ab - ac - bc) / 9(a - b)(b - c)

** (a² + b² + c² - ab - ac - bc) / 3(a - b)(b - c)**

Conclusion

Therefore, the simplified form of the expression (a-b)^3 + (b-c)^3 + (c-a)^3 / 9(a-b)(b-c)(c-a) is (a² + b² + c² - ab - ac - bc) / 3(a - b)(b - c).