%5E5-binomial-theorem.jpeg "(x+y)^5 Binomial Theorem")
The Binomial Theorem: Expanding (x+y)^5
The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form (x+y)^n, where n is a positive integer. In this article, we will focus on expanding (x+y)^5 using the binomial theorem.
The Binomial Theorem Formula
The binomial theorem formula is given by:
(x+y)^n = ∑[from k=0 to n] (n choose k) * x^(n-k) * y^k
where (n choose k) is the binomial coefficient, which can be calculated using the formula:
(n choose k) = n! / (k!(n-k)!)
Expanding (x+y)^5
To expand (x+y)^5, we can use the binomial theorem formula with n=5. We get:
(x+y)^5 = ∑[from k=0 to 5] (5 choose k) * x^(5-k) * y^k
Using the formula for the binomial coefficient, we can calculate the values of (5 choose k) as follows:
- (5 choose 0) = 5! / (0!(5-0)!) = 1
- (5 choose 1) = 5! / (1!(5-1)!) = 5
- (5 choose 2) = 5! / (2!(5-2)!) = 10
- (5 choose 3) = 5! / (3!(5-3)!) = 10
- (5 choose 4) = 5! / (4!(5-4)!) = 5
- (5 choose 5) = 5! / (5!(5-5)!) = 1
Now, we can plug these values into the binomial theorem formula:
(x+y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5
Simplifying the Expansion
We can simplify the expansion by combining like terms:
(x+y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5
This is the final expansion of (x+y)^5 using the binomial theorem.
Conclusion
The binomial theorem is a powerful tool for expanding expressions of the form (x+y)^n. By using the formula and calculating the binomial coefficients, we can expand (x+y)^5 to get a simplified expression. This expansion can be useful in a variety of mathematical and real-world applications.
