%5E3-expand.jpeg "(a+b)^3 Expand")
Expanding (a+b)^3
The expansion of (a+b)^3 is a fundamental concept in algebra, often used in various mathematical applications. Understanding its expansion is crucial for simplifying expressions, solving equations, and performing other algebraic operations.
Understanding the Expansion
The expansion of (a+b)^3 represents the product of (a+b) multiplied by itself three times:
(a+b)^3 = (a+b)(a+b)(a+b)
To expand this, we can apply the distributive property multiple times:
- First expansion: (a+b)(a+b) = a(a+b) + b(a+b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2
- Second expansion: (a^2 + 2ab + b^2)(a+b) = a(a^2 + 2ab + b^2) + b(a^2 + 2ab + b^2) = a^3 + 2a^2b + ab^2 + ba^2 + 2ab^2 + b^3 = a^3 + 3a^2b + 3ab^2 + b^3
Therefore, the complete expansion of (a+b)^3 is:
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
Applying the Binomial Theorem
The expansion of (a+b)^3 can also be derived using the Binomial Theorem, which provides a general formula for expanding any binomial raised to a power. The binomial theorem states:
(a+b)^n = ∑(n choose k) a^(n-k) b^k
where:
- n is the power to which the binomial is raised
- k is a non-negative integer ranging from 0 to n
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
For (a+b)^3, we have n = 3. Applying the binomial theorem:
(a+b)^3 = (3 choose 0)a^3b^0 + (3 choose 1)a^2b^1 + (3 choose 2)a^1b^2 + (3 choose 3)a^0b^3
Calculating the binomial coefficients:
- (3 choose 0) = 3! / (0! * 3!) = 1
- (3 choose 1) = 3! / (1! * 2!) = 3
- (3 choose 2) = 3! / (2! * 1!) = 3
- (3 choose 3) = 3! / (3! * 0!) = 1
Substituting the coefficients back into the expansion:
(a+b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3 = a^3 + 3a^2b + 3ab^2 + b^3
Conclusion
Expanding (a+b)^3 results in the expression a^3 + 3a^2b + 3ab^2 + b^3. This expansion can be obtained through repeated application of the distributive property or by utilizing the Binomial Theorem. Understanding this expansion is essential for various algebraic manipulations and applications.