%5E6-binomial-expansion.jpeg "(x+1/x)^6 Binomial Expansion")
Binomial Expansion of (x+1/x)^6
In mathematics, binomial expansion is a process of expanding an expression of the form (a+b)^n, where a and b are variables and n is a positive integer. In this article, we will explore the binomial expansion of (x+1/x)^6.
What is Binomial Expansion?
Binomial expansion is a method of expanding a binomial expression into a sum of terms involving various powers of the variables. The general formula for binomial expansion is:
(a+b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + ... + b^n
where n is a positive integer.
Expanding (x+1/x)^6
Using the binomial expansion formula, we can expand (x+1/x)^6 as follows:
(x+1/x)^6 = x^6 + 6x^5(1/x) + 15x^4(1/x)^2 + 20x^3(1/x)^3 + 15x^2(1/x)^4 + 6x(1/x)^5 + (1/x)^6
Simplifying the expression, we get:
(x+1/x)^6 = x^6 + 6x^4 + 15x^2 + 20 + 15/x^2 + 6/x^4 + 1/x^6
Simplifying the Expression
We can further simplify the expression by combining like terms:
(x+1/x)^6 = x^6 + 6x^4 + 15x^2 + 20 + 15/x^2 + 6/x^4 + 1/x^6
Conclusion
In this article, we have successfully expanded (x+1/x)^6 using binomial expansion. The final expression is a sum of seven terms involving various powers of x. This expansion is useful in various mathematical and scientific applications.
Example Problems
- Expand
(x+2/x)^3using binomial expansion. - Simplify the expression
(x+1/x)^4using binomial expansion.
References
- Binomial Expansion Formula:
- Binomial Expansion Examples:
Note: This article is for educational purposes only and should not be used for commercial purposes.
