## (a + b + c)³ Formula Proof

The formula for expanding (a + b + c)³ is a fundamental concept in algebra. This formula allows us to expand a cubed expression into a sum of individual terms.

### Formula:

**(a + b + c)³ = a³ + b³ + c³ + 3a²b + 3a²c + 3ab² + 3ac² + 3b²c + 3bc² + 6abc**

### Proof:

We can prove this formula by using the distributive property of multiplication and by expanding the expression step by step.

**Step 1: Expand (a + b + c)³ as (a + b + c)(a + b + c)(a + b + c)**

**Step 2: Expand the first two factors using the distributive property:**

(a + b + c)(a + b + c) = a(a + b + c) + b(a + b + c) + c(a + b + c) = a² + ab + ac + ba + b² + bc + ca + cb + c² = a² + b² + c² + 2ab + 2ac + 2bc

**Step 3: Multiply the result from Step 2 by (a + b + c):**

(a² + b² + c² + 2ab + 2ac + 2bc)(a + b + c) = a(a² + b² + c² + 2ab + 2ac + 2bc) + b(a² + b² + c² + 2ab + 2ac + 2bc) + c(a² + b² + c² + 2ab + 2ac + 2bc)

**Step 4: Expand the above expression:**

= a³ + ab² + ac² + 2a²b + 2a²c + 2abc + ba² + b³ + bc² + 2ab² + 2abc + 2b²c + ca² + cb² + c³ + 2abc + 2ac² + 2bc²

**Step 5: Combine like terms:**

= **a³ + b³ + c³ + 3a²b + 3a²c + 3ab² + 3ac² + 3b²c + 3bc² + 6abc**

**Therefore, we have proved the formula (a + b + c)³ = a³ + b³ + c³ + 3a²b + 3a²c + 3ab² + 3ac² + 3b²c + 3bc² + 6abc**

### Applications:

This formula is widely used in various mathematical fields, including:

**Algebraic simplification:**Expanding expressions involving (a + b + c)³**Polynomial factorization:**Factoring expressions involving cubes**Solving equations:**Finding solutions to equations involving cubed terms**Calculus:**Calculating derivatives and integrals involving cubic functions

The formula for (a + b + c)³ is a valuable tool for simplifying complex expressions and solving problems involving cubic equations.