%5E3%2B(b%2Bc)%5E3%2B(c%2Ba)%5E3-3(a%2Bb)(b%2Bc)(c%2Ba)%3D2(a%5E3%2Bb%5E3%2Bc%5E3-3abc).jpeg "(a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)=2(a^3+b^3+c^3-3abc)")
The Expansion and Proof of (a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)=2(a^3+b^3+c^3-3abc)
This algebraic identity presents a fascinating relationship between cubic expressions. Let's break it down:
Understanding the Identity
The identity states that the expression:
(a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)
is equivalent to:
2(a^3+b^3+c^3-3abc)
Proof by Expansion
We can prove this identity by expanding the expressions and simplifying:
-
Expand the cubes:
- (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
- (b+c)^3 = b^3 + 3b^2c + 3bc^2 + c^3
- (c+a)^3 = c^3 + 3c^2a + 3ca^2 + a^3
-
Substitute the expanded terms into the original expression:
- (a^3 + 3a^2b + 3ab^2 + b^3) + (b^3 + 3b^2c + 3bc^2 + c^3) + (c^3 + 3c^2a + 3ca^2 + a^3) - 3(a+b)(b+c)(c+a)
-
Expand the product (a+b)(b+c)(c+a):
- (a+b)(b+c)(c+a) = abc + a^2b + ab^2 + ac^2 + a^2c + abc + b^2c + bc^2
-
Simplify the expression by combining like terms:
- 2(a^3 + b^3 + c^3) + 3(a^2b + ab^2 + b^2c + bc^2 + c^2a + ca^2) - 3(abc + a^2b + ab^2 + ac^2 + a^2c + abc + b^2c + bc^2)
-
Cancel out terms:
- 2(a^3 + b^3 + c^3) - 6abc
-
Factor out 2:
- 2(a^3 + b^3 + c^3 - 3abc)
Therefore, we have proven that (a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)=2(a^3+b^3+c^3-3abc)
Applications
This identity has applications in various areas of mathematics, including:
- Algebraic manipulation: It can be used to simplify complex expressions and solve equations.
- Number theory: It helps to understand relationships between numbers and their cubes.
- Geometry: It can be used to derive geometric formulas and solve problems related to volumes and surface areas.
The identity (a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)=2(a^3+b^3+c^3-3abc) demonstrates the beauty and elegance of algebraic relationships. It serves as a valuable tool for mathematicians and anyone working with algebraic expressions.