%5E3-proof.jpeg "(a+b)^3 Proof")
The Proof of (a + b)³
The expansion of (a + b)³ is a fundamental concept in algebra. It is often used in various mathematical contexts and can be derived using two main methods:
1. Using the Distributive Property
This method involves repeatedly applying the distributive property of multiplication over addition:
Step 1: Expand the cube of the binomial. (a + b)³ = (a + b) * (a + b) * (a + b)
Step 2: Expand the first two factors using the distributive property. (a + b) * (a + b) = a(a + b) + b(a + b) = a² + ab + ba + b² = a² + 2ab + b²
Step 3: Multiply the result from Step 2 by (a + b). (a² + 2ab + b²) * (a + b) = a(a² + 2ab + b²) + b(a² + 2ab + b²)
Step 4: Apply the distributive property again. a(a² + 2ab + b²) + b(a² + 2ab + b²) = a³ + 2a²b + ab² + ba² + 2ab² + b³
Step 5: Combine like terms. a³ + 2a²b + ab² + ba² + 2ab² + b³ = a³ + 3a²b + 3ab² + b³
Therefore, (a + b)³ = a³ + 3a²b + 3ab² + b³
2. Using the Binomial Theorem
The binomial theorem provides a general formula for expanding any binomial raised to a power. It states:
(a + b)ⁿ = ∑(k=0 to n) (n choose k) * a^(n-k) * b^k
where (n choose k) represents the binomial coefficient, calculated as n!/(k!(n-k)!).
Step 1: Apply the binomial theorem to (a + b)³.
(a + b)³ = ∑(k=0 to 3) (3 choose k) * a^(3-k) * b^k
Step 2: Expand the summation.
(a + b)³ = (3 choose 0) * a³ * b⁰ + (3 choose 1) * a² * b¹ + (3 choose 2) * a¹ * b² + (3 choose 3) * a⁰ * b³
Step 3: Calculate the binomial coefficients.
(3 choose 0) = 1, (3 choose 1) = 3, (3 choose 2) = 3, (3 choose 3) = 1
Step 4: Substitute the coefficients and simplify.
(a + b)³ = 1 * a³ * 1 + 3 * a² * b + 3 * a * b² + 1 * 1 * b³ = a³ + 3a²b + 3ab² + b³
Therefore, using both methods, we have shown that:
(a + b)³ = a³ + 3a²b + 3ab² + b³
This expansion is useful for various algebraic manipulations, and it forms the basis for further expansions of higher powers of binomials.