%5E5-binomial-expansion.jpeg "(x+y)^5 Binomial Expansion")
Binomial Expansion of (x+y)^5
In algebra, the binomial expansion of (x+y)^n is a fundamental concept that represents the expression of a binomial raised to a power n, where n is a positive integer. In this article, we will focus on the binomial expansion of (x+y)^5.
The Binomial Theorem
The binomial theorem states that for any positive integer n,
$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$
where $\binom{n}{k}$ is the binomial coefficient, which can be calculated as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Binomial Expansion of (x+y)^5
Using the binomial theorem, we can expand (x+y)^5 as follows:
$(x+y)^5 = \sum_{k=0}^{5} \binom{5}{k} x^{5-k} y^k$
Expanding the summation, we get:
$(x+y)^5 = \binom{5}{0} x^5 y^0 + \binom{5}{1} x^4 y^1 + \binom{5}{2} x^3 y^2 + \binom{5}{3} x^2 y^3 + \binom{5}{4} x^1 y^4 + \binom{5}{5} x^0 y^5$
Simplifying the expression, we get:
$(x+y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5$
Conclusion
In conclusion, the binomial expansion of (x+y)^5 is a powerful tool for expanding expressions of the form (x+y)^n. By using the binomial theorem, we can easily expand (x+y)^5 to obtain the expression x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5. This expansion has numerous applications in algebra, calculus, and other areas of mathematics.
