%5E7-binomial-expansion.jpeg "(x+y)^7 Binomial Expansion")
Binomial Expansion of (x+y)^7
Introduction
Binomial expansion is a fundamental concept in algebra, which involves expanding an expression of the form $(a+b)^n$, where $a$ and $b$ are variables and $n$ is a positive integer. One of the most popular formulas in binomial expansion is the binomial theorem, which states that:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where $\binom{n}{k}$ is the binomial coefficient, also known as the "choose" function, which represents the number of ways to choose $k$ items from a set of $n$ items.
Binomial Expansion of (x+y)^7
In this article, we will explore the binomial expansion of $(x+y)^7$, which is a special case of the binomial theorem.
Using the binomial theorem, we can expand $(x+y)^7$ as follows:
$(x+y)^7 = \sum_{k=0}^7 \binom{7}{k} x^{7-k} y^k$
Expanding the Binomial Coefficients
To expand the binomial coefficients, we need to calculate the values of $\binom{7}{k}$ for $k=0, 1, 2, ..., 7$.
| k | binom(7, k) |
|---|---|
| 0 | 1 |
| 1 | 7 |
| 2 | 21 |
| 3 | 35 |
| 4 | 35 |
| 5 | 21 |
| 6 | 7 |
| 7 | 1 |
The Expanded Form
Now, we can plug in the values of the binomial coefficients into the expansion formula:
$(x+y)^7 = 1x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + 1y^7$
Conclusion
In this article, we have derived the binomial expansion of $(x+y)^7$ using the binomial theorem. The expanded form of the expression is a polynomial of degree 7, with coefficients that can be calculated using the binomial coefficients. This expansion has many applications in mathematics, physics, and engineering, particularly in problems involving combinatorics and probability.
I hope this article helps you understand the binomial expansion of $(x+y)^7$!
