%5E5-binomial-expansion.jpeg "(2x-3y)^5 Binomial Expansion")
Binomial Expansion of (2x-3y)^5
Introduction
In mathematics, binomial expansion is a technique used to expand an expression of the form (a+b)^n, where a and b are variables and n is a non-negative integer. This technique is widely used in algebra, calculus, and other branches of mathematics. In this article, we will explore the binomial expansion of (2x-3y)^5.
Binomial Theorem
The binomial theorem states that for any positive integer n,
$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$
where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.
Expanding (2x-3y)^5
Using the binomial theorem, we can expand (2x-3y)^5 as follows:
$(2x-3y)^5 = \sum_{k=0}^{5} \binom{5}{k} (2x)^{5-k} (-3y)^k$
Simplifying the expression, we get:
$(2x-3y)^5 = \binom{5}{0} (2x)^5 + \binom{5}{1} (2x)^4 (-3y) + \binom{5}{2} (2x)^3 (-3y)^2 + \binom{5}{3} (2x)^2 (-3y)^3 + \binom{5}{4} (2x) (-3y)^4 + \binom{5}{5} (-3y)^5$
Evaluating the binomial coefficients, we get:
$(2x-3y)^5 = 32x^5 - 240x^4y + 720x^3y^2 - 1080x^2y^3 + 810xy^4 - 243y^5$
Conclusion
In this article, we have explored the binomial expansion of (2x-3y)^5 using the binomial theorem. The expansion is a polynomial of degree 5 in terms of x and y, with coefficients obtained using the binomial coefficients. The resulting expression is a useful tool in various mathematical applications, such as algebra, calculus, and geometry.
