%5E3-formula-proof.jpeg "(a+b+c)^3 Formula Proof")
(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.