%5E4-binomial-expansion.jpeg "(x+2y)^4 Binomial Expansion")
Binomial Expansion of (x+2y)^4
In algebra, the binomial theorem is a powerful tool for expanding powers of a binomial expression. A binomial expression is a polynomial expression consisting of two terms, such as (x+2y). In this article, we will explore the binomial expansion of (x+2y)^4.
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:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Expanding (x+2y)^4
Using the binomial theorem, we can expand (x+2y)^4 as follows:
$(x+2y)^4 = \sum_{k=0}^4 \binom{4}{k} x^{4-k} (2y)^k$
Let's calculate each term:
Term 1: k=0
$\binom{4}{0} x^{4-0} (2y)^0 = 1 \cdot x^4 \cdot 1 = x^4$
Term 2: k=1
$\binom{4}{1} x^{4-1} (2y)^1 = 4 \cdot x^3 \cdot 2y = 8x^3y$
Term 3: k=2
$\binom{4}{2} x^{4-2} (2y)^2 = 6 \cdot x^2 \cdot 4y^2 = 24x^2y^2$
Term 4: k=3
$\binom{4}{3} x^{4-3} (2y)^3 = 4 \cdot x \cdot 8y^3 = 32xy^3$
Term 5: k=4
$\binom{4}{4} x^{4-4} (2y)^4 = 1 \cdot x^0 \cdot 16y^4 = 16y^4$
Final Expansion
Combining all the terms, we get the binomial expansion of (x+2y)^4:
$(x+2y)^4 = x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4$
This expansion can be used to simplify expressions, solve equations, and perform other algebraic manipulations.
