%5E6-binomial-theorem.jpeg "(x+1/x)^6 Binomial Theorem")
Binomial Theorem: Expanding (x+1/x)^6
The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form (a+b)^n, where a and b are variables and n is a positive integer. In this article, we will explore how to use the binomial theorem to expand the expression (x+1/x)^6.
The Binomial Theorem Formula
The binomial theorem formula is given by:
(a+b)^n = a^n + na^(n-1)b + (n(n-1)/2!)a^(n-2)b^2 + ... + b^n
where n is a positive integer.
Expanding (x+1/x)^6
To expand (x+1/x)^6, we can use the binomial theorem formula by substituting a=x and b=1/x. We get:
(x+1/x)^6 = x^6 + 6x^5(1/x) + (6(5)/2!)x^4(1/x)^2 + (6(5)(4)/3!)x^3(1/x)^3 + (6(5)(4)(3)/4!)x^2(1/x)^4 + (6(5)(4)(3)(2)/5!)x(1/x)^5 + (1/x)^6
Simplifying the Expansion
Now, we can simplify the expansion 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 used the binomial theorem to expand the expression (x+1/x)^6. We have seen how the binomial theorem formula can be applied to simplify the expansion and arrive at the final result.
Important Notes
- The binomial theorem can be used to expand expressions of the form (a+b)^n, where a and b are variables and n is a positive integer.
- The binomial theorem is a powerful tool in algebra and has many applications in mathematics, physics, and engineering.
- The expansion of (x+1/x)^6 can be used to solve problems in calculus, algebra, and other areas of mathematics.
