%5E4-binomial-expansion.jpeg "(2x+3y)^4 Binomial Expansion")
Binomial Expansion of (2x+3y)^4
In mathematics, binomial expansion is a useful technique to expand powers of a binomial expression. A binomial expression is an algebraic expression consisting of two terms, such as (2x+3y). In this article, we will explore the binomial expansion of (2x+3y)^4.
Understanding Binomial Expansion
Before we dive into the expansion of (2x+3y)^4, let's recall the basic concept of binomial expansion. The binomial expansion of (a+b)^n, where 'a' and 'b' are constants and 'n' is a positive integer, is given by:
(a+b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + ... + n(n-1)(n-2)...2a^1b^(n-1) + b^n
This formula is known as the Binomial Theorem.
Expanding (2x+3y)^4
Now, let's apply the Binomial Theorem to expand (2x+3y)^4. Here, a = 2x and b = 3y, and n = 4.
(2x+3y)^4 = (2x)^4 + 4(2x)^3(3y) + 6(2x)^2(3y)^2 + 4(2x)(3y)^3 + (3y)^4
Simplifying the expression, we get:
(2x+3y)^4 = 16x^4 + 96x^3y + 216x^2y^2 + 216xy^3 + 81y^4
Conclusion
In this article, we have successfully expanded the binomial expression (2x+3y)^4 using the Binomial Theorem. The resulting expression is a polynomial of degree 4, consisting of five terms. This expansion can be useful in various mathematical applications, such as algebra, calculus, and geometry.
Remember
When expanding a binomial expression, it's essential to remember the formula for the Binomial Theorem and to simplify the expression carefully to avoid any errors. With practice, you'll become proficient in expanding binomial expressions of any degree!
