The (a + b)^5 Formula: Expanding Binomial Powers
The formula for (a + b)^5 is a special case of the Binomial Theorem, which provides a general way to expand expressions of the form (x + y)^n.
Here's how to expand (a + b)^5:
(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
Understanding the Formula:
- Coefficients: The coefficients in the expansion (1, 5, 10, 10, 5, 1) are determined by Pascal's Triangle. Each row of Pascal's Triangle represents the coefficients for a binomial expansion of a specific power.
- Exponents: The exponents of 'a' decrease from 5 to 0, while the exponents of 'b' increase from 0 to 5.
- Symmetry: The coefficients and exponents follow a symmetrical pattern.
How to Derive the Formula:
You can derive the formula by repeated multiplication:
- (a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2
- (a + b)^3 = (a + b)^2(a + b) = (a^2 + 2ab + b^2)(a + b) = a^3 + 3a^2b + 3ab^2 + b^3
- (a + b)^4 = (a + b)^3(a + b) = (a^3 + 3a^2b + 3ab^2 + b^3)(a + b) = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
- (a + b)^5 = (a + b)^4(a + b) = (a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4)(a + b) = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
Example:
Let's expand (2x + 3y)^5 using the formula:
(2x + 3y)^5 = (2x)^5 + 5(2x)^4(3y) + 10(2x)^3(3y)^2 + 10(2x)^2(3y)^3 + 5(2x)(3y)^4 + (3y)^5
Simplifying:
(2x + 3y)^5 = 32x^5 + 240x^4y + 720x^3y^2 + 1080x^2y^3 + 810xy^4 + 243y^5
Conclusion:
The (a + b)^5 formula is a powerful tool for expanding binomial expressions. Understanding the formula and its derivation can be helpful for various mathematical problems and applications.
