%5E3-derivation.jpeg "(a+b)^3 Derivation")
Deriving the Cube of a Binomial: (a + b)³
The expansion of (a + b)³ is a fundamental concept in algebra, frequently used in various mathematical disciplines. Here's a step-by-step derivation of this formula:
1. Expanding the Expression
We can write (a + b)³ as the product of three identical binomials:
(a + b)³ = (a + b)(a + b)(a + b)
2. Applying the Distributive Property
Let's start by expanding the first two binomials:
(a + b)(a + b) = a(a + b) + b(a + b)
Using the distributive property, we get:
(a + b)(a + b) = a² + ab + ba + b²
Since multiplication is commutative, ab = ba, so we can simplify this to:
(a + b)(a + b) = a² + 2ab + b²
3. Expanding the Entire Expression
Now we need to multiply this result by the remaining (a + b):
(a + b)³ = (a² + 2ab + b²)(a + b)
Applying the distributive property once more:
(a + b)³ = a²(a + b) + 2ab(a + b) + b²(a + b)
Expanding each term:
(a + b)³ = a³ + a²b + 2a²b + 2ab² + b²a + b³
4. Combining Like Terms
Finally, we combine the terms with the same variables and exponents:
(a + b)³ = a³ + 3a²b + 3ab² + b³
Conclusion
Therefore, the expansion of (a + b)³ is:
(a + b)³ = a³ + 3a²b + 3ab² + b³
This formula is useful for expanding expressions, simplifying equations, and understanding other algebraic concepts.