%5E3%2B(b%2Bc)%5E3%2B(c%2Ba)%5E3-3(a%2Bb)(b%2Bc)(c%2Ba).jpeg "(a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)")
Factoring the Expression (a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)
This expression might look daunting at first glance, but it's a classic example of a factorization pattern that arises frequently in algebra. Let's break it down step-by-step.
The Key Identity
The core of the factorization lies in the following identity:
x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - xz - yz)
Applying the Identity
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Recognize the Structure: Notice that our original expression has the same form as the left-hand side of the identity. Let's make the following substitutions:
- x = a + b
- y = b + c
- z = c + a
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Substitute and Simplify: Substituting these values into the identity, we get:
(a + b)³ + (b + c)³ + (c + a)³ - 3(a + b)(b + c)(c + a) = [(a + b) + (b + c) + (c + a)][(a + b)² + (b + c)² + (c + a)² - (a + b)(b + c) - (a + b)(c + a) - (b + c)(c + a)]
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Expand and Combine: Now, let's expand the squares and simplify:
= (2a + 2b + 2c)[(a² + 2ab + b²) + (b² + 2bc + c²) + (c² + 2ac + a²) - (ab + ac + b² + bc) - (ac + a² + bc + ab) - (bc + b² + ac + c²)]
= 2(a + b + c)[a² + b² + c² - ab - ac - bc]
The Final Result
Therefore, the factored form of the expression is:
(a + b)³ + (b + c)³ + (c + a)³ - 3(a + b)(b + c)(c + a) = 2(a + b + c)(a² + b² + c² - ab - ac - bc)
This factorization is useful in various contexts, including solving equations, simplifying expressions, and understanding relationships between variables.