5-binomial-theorem.jpeg "(2x+3y)5 Binomial Theorem")
The Binomial Theorem: Expanding (2x+3y)^5
The binomial theorem is a powerful tool in algebra that allows us to expand powers of binomials, such as (2x+3y)^5, into a polynomial expression. In this article, we will explore the binomial theorem and use it to expand the given expression.
What is the Binomial Theorem?
The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial, which is an expression consisting of two terms. The theorem states that:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where a and b are the two terms of the binomial, n is a positive integer, and $\binom{n}{k}$ is the binomial coefficient, given by:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Expanding (2x+3y)^5
Using the binomial theorem, we can expand (2x+3y)^5 as follows:
$(2x+3y)^5 = \sum_{k=0}^5 \binom{5}{k} (2x)^{5-k} (3y)^k$
Now, let's calculate the binomial coefficients and expand the expression:
$(2x+3y)^5 = \binom{5}{0} (2x)^5 (3y)^0 + \binom{5}{1} (2x)^4 (3y)^1 + \binom{5}{2} (2x)^3 (3y)^2 + \binom{5}{3} (2x)^2 (3y)^3 + \binom{5}{4} (2x)^1 (3y)^4 + \binom{5}{5} (2x)^0 (3y)^5$
$= 32x^5 + 5(16x^4)(3y) + 10(8x^3)(9y^2) + 10(4x^2)(27y^3) + 5(2x)(81y^4) + 243y^5$
$= 32x^5 + 240x^4y + 720x^3y^2 + 1080x^2y^3 + 810xy^4 + 243y^5$
And there you have it! The expansion of (2x+3y)^5 using the binomial theorem.
Conclusion
In this article, we have seen how the binomial theorem can be used to expand powers of binomials, such as (2x+3y)^5. The theorem provides a powerful tool for algebraic manipulations and is widely used in various branches of mathematics and science.
