%5E6-binomial-expansion.jpeg "(1+x)^6 Binomial Expansion")
Binomial Expansion of (1+x)^6
The binomial theorem is a fundamental concept in algebra that allows us to expand powers of a binomial expression, which is an expression consisting of two terms. In this article, we will explore the binomial expansion of (1+x)^6.
What is Binomial Expansion?
Binomial expansion is a method of expanding powers of a binomial expression into a sum of terms involving various powers of the individual terms. The general formula for binomial expansion is:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where a and b are the two terms of the binomial expression, n is a positive integer, and \binom{n}{k} is the binomial coefficient.
Binomial Expansion of (1+x)^6
Using the binomial theorem, we can expand (1+x)^6 as follows:
$(1+x)^6 = \sum_{k=0}^6 \binom{6}{k} 1^{6-k} x^k$
Simplifying the expression, we get:
$(1+x)^6 = \binom{6}{0} 1^6 x^0 + \binom{6}{1} 1^5 x^1 + \binom{6}{2} 1^4 x^2 + \binom{6}{3} 1^3 x^3 + \binom{6}{4} 1^2 x^4 + \binom{6}{5} 1^1 x^5 + \binom{6}{6} 1^0 x^6$
Evaluating the binomial coefficients, we get:
$(1+x)^6 = 1 + 6x + 15x^2 + 20x^3 + 15x^4 + 6x^5 + x^6$
Simplified Form
The binomial expansion of (1+x)^6 can be written in a more compact form as:
$(1+x)^6 = 1 + 6x + 15x^2 + 20x^3 + 15x^4 + 6x^5 + x^6$
Conclusion
In this article, we have explored the binomial expansion of (1+x)^6. We have seen how to use the binomial theorem to expand powers of a binomial expression and simplify the result. The binomial expansion of (1+x)^6 has many applications in mathematics, physics, and engineering, and is an important concept to understand in algebra and calculus.
