## The Algebraic Identity: (a+b)^3 + (a-b)^3 = 2a(a^2 + 3b^2)

This article explores the algebraic identity (a+b)^3 + (a-b)^3 = 2a(a^2 + 3b^2), demonstrating how it's derived and its applications.

### Derivation of the Identity

To prove the identity, we can expand the cubes on the left-hand side using the binomial theorem:

**(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3****(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3**

Now, we add these two expansions:

**(a+b)^3 + (a-b)^3 = (a^3 + 3a^2b + 3ab^2 + b^3) + (a^3 - 3a^2b + 3ab^2 - b^3)**

Notice that the terms with odd powers of *b* cancel out, leaving us with:

**(a+b)^3 + (a-b)^3 = 2a^3 + 6ab^2**

Factoring out 2a gives us the right-hand side of the identity:

**(a+b)^3 + (a-b)^3 = 2a(a^2 + 3b^2)**

### Applications of the Identity

This identity can be used to simplify expressions, solve equations, and prove other identities. Here are some examples:

**Simplifying expressions:**If we're given an expression of the form (a+b)^3 + (a-b)^3, we can immediately use the identity to simplify it to 2a(a^2 + 3b^2).**Solving equations:**If we have an equation like (x+2)^3 + (x-2)^3 = 8, we can use the identity to simplify the equation to 2x(x^2 + 12) = 8. This simplified equation can then be solved for x.**Proving other identities:**The identity can be used as a stepping stone to prove other identities. For example, we can use it to prove the identity (a+b)^3 - (a-b)^3 = 6ab(a+b).

### Conclusion

The algebraic identity (a+b)^3 + (a-b)^3 = 2a(a^2 + 3b^2) is a useful tool for simplifying expressions, solving equations, and proving other identities. Understanding its derivation and applications can enhance your understanding of algebraic manipulations and problem-solving techniques.