5-binomial-theorem.jpeg "(x+y)5 Binomial Theorem")
(x+y)^5: Understanding the Binomial Theorem
The binomial theorem is a fundamental concept in algebra that allows us to expand powers of binomials, which are expressions consisting of two terms. In this article, we will explore the expansion of (x+y)^5 using the binomial theorem.
What is the Binomial Theorem?
The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial. It states that:
$(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$
where n is a positive integer, and k is an integer that ranges from 0 to n. The symbol \binom{n}{k} represents the number of combinations of n items taken k at a time, also known as the binomial coefficient.
Expanding (x+y)^5
To expand (x+y)^5, we can use the binomial theorem by substituting n=5 into the formula. This gives us:
$(x+y)^5 = \sum_{k=0}^5 \binom{5}{k} x^{5-k} y^k$
Now, let's compute the binomial coefficients \binom{5}{k} for k=0, 1, 2, 3, 4, 5:
\binom{5}{0} = 1\binom{5}{1} = 5\binom{5}{2} = 10\binom{5}{3} = 10\binom{5}{4} = 5\binom{5}{5} = 1
Substituting these values into the formula, we get:
$(x+y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5$
This is the expanded form of (x+y)^5.
Conclusion
In this article, we have explored the expansion of (x+y)^5 using the binomial theorem. By understanding the binomial theorem and its application, we can easily expand powers of binomials and simplify algebraic expressions. The expansion of (x+y)^5 is a fundamental result in algebra that has numerous applications in mathematics, physics, and other fields.
