5-formula.jpeg "(a+b)5 Formula")
(a+b)^5 Formula: Understanding the Binomial Expansion
The (a+b)^5 formula is a fundamental concept in algebra, used to expand the power of a binomial expression. In this article, we will explore the formula, its derivation, and some examples to illustrate its application.
What is the (a+b)^5 Formula?
The (a+b)^5 formula is a special case of the binomial theorem, which states that:
(a+b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
This formula allows us to expand the power of a binomial expression (a+b) raised to the fifth power.
Derivation of the (a+b)^5 Formula
To derive the (a+b)^5 formula, we can use the binomial theorem, which states that:
(a+b)^n = a^n + na^(n-1)b + (n(n-1))/2! a^(n-2)b^2 + … + b^n
where n is a positive integer.
By substituting n=5 into the binomial theorem, we get:
(a+b)^5 = a^5 + 5a^4b + (5(4))/2! a^3b^2 + (5(4)(3))/3! a^2b^3 + (5(4)(3)(2))/4! ab^4 + b^5
Simplifying the expression, we get:
(a+b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
Examples and Applications
Here are some examples to illustrate the application of the (a+b)^5 formula:
Example 1: Expand (x+2)^5
Using the (a+b)^5 formula, we get:
(x+2)^5 = x^5 + 5x^4(2) + 10x^3(2^2) + 10x^2(2^3) + 5x(2^4) + 2^5
= x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32
Example 2: Expand (3y-2)^5
Using the (a+b)^5 formula, we get:
(3y-2)^5 = (3y)^5 - 5(3y)^4(2) + 10(3y)^3(2^2) - 10(3y)^2(2^3) + 5(3y)(2^4) - 2^5
= 243y^5 - 810y^4 + 1080y^3 - 720y^2 + 180y - 32
Conclusion
The (a+b)^5 formula is a powerful tool for expanding binomial expressions raised to the fifth power. It has numerous applications in algebra, calculus, and other branches of mathematics. By understanding the derivation and application of this formula, you can simplify complex expressions and solve problems more efficiently.
