(x+y)^6 Binomial Expansion

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Jul 03, 2024

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Binomial Expansion of (x+y)^6

In algebra, the binomial expansion of a power of a binomial expression is a fundamental concept. In this article, we will explore the binomial expansion of (x+y)^6.

What is Binomial Expansion?

The binomial expansion is a method of expanding a power of a binomial expression into a sum of terms involving various powers of the individual variables. The general formula for the binomial expansion of (x+y)^n is:

(x+y)^n = x^n + nx^(n-1)y + n(n-1)x^(n-2)y^2 + ... + ny^(n-1)x + y^n

where n is a positive integer.

Binomial Expansion of (x+y)^6

Using the general formula, we can expand (x+y)^6 as follows:

(x+y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6

To understand how we arrived at this expansion, let's break it down:

• The first term is x^6, which is the sixth power of x.
• The second term is 6x^5y, which is the product of x raised to the power of 5 and y raised to the power of 1. The coefficient 6 is calculated using the formula for combinations: 6 = 6! / (5! \* 1!).
• The third term is 15x^4y^2, which is the product of x raised to the power of 4 and y raised to the power of 2. The coefficient 15 is calculated using the formula for combinations: 15 = 6! / (4! \* 2!).
• We continue this process until we reach the last term, y^6, which is the sixth power of y.

Applications of Binomial Expansion

The binomial expansion has many applications in mathematics, science, and engineering. Some examples include:

• Algebra: Binomial expansion is used to simplify algebraic expressions and solve equations.
• Calculus: Binomial expansion is used to find the derivatives and integrals of functions.
• Statistics: Binomial expansion is used to calculate probabilities in statistics and data analysis.
• Physics: Binomial expansion is used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.

In conclusion, the binomial expansion of (x+y)^6 is a powerful tool for simplifying and solving algebraic expressions. By understanding the general formula and applying it to specific cases, we can unlock a wide range of applications in mathematics and beyond.