(a+b)^3 Expanded

Jasonbradley

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Jun 16, 2024

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Expanding (a + b)^3

The expansion of (a + b)^3 is a fundamental concept in algebra, and understanding it is crucial for various mathematical operations. Let's explore how to expand this expression and the different methods used.

Using the Distributive Property

The most straightforward method to expand (a + b)^3 is by using the distributive property multiple times.

1. First Expansion: (a + b)^3 = (a + b)(a + b)(a + b)

2. Second Expansion: (a + b)(a + b) = a^2 + 2ab + b^2

3. Third Expansion: (a + b)(a^2 + 2ab + b^2) = a(a^2 + 2ab + b^2) + b(a^2 + 2ab + b^2)

4. Simplifying: a^3 + 2a^2b + ab^2 + a^2b + 2ab^2 + b^3 = a^3 + 3a^2b + 3ab^2 + b^3

Using the Binomial Theorem

The binomial theorem provides a general formula for expanding any expression of the form (x + y)^n.

The formula states:

(x + y)^n = ∑ (n choose k) x^(n-k) y^k

where (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).

For (a + b)^3, n = 3. Let's apply the formula:

1. k = 0: (3 choose 0) a^(3-0) b^0 = 1 * a^3 * 1 = a^3
2. k = 1: (3 choose 1) a^(3-1) b^1 = 3 * a^2 * b = 3a^2b
3. k = 2: (3 choose 2) a^(3-2) b^2 = 3 * a * b^2 = 3ab^2
4. k = 3: (3 choose 3) a^(3-3) b^3 = 1 * 1 * b^3 = b^3

Therefore, the expansion of (a + b)^3 using the binomial theorem is:

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Conclusion

Both methods lead to the same result:

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Understanding the expansion of (a + b)^3 is essential for various algebraic manipulations, including solving equations, simplifying expressions, and working with polynomial functions.