%5E6-binomial-expansion.jpeg "(x+1)^6 Binomial Expansion")
Binomial Expansion of (x+1)^6
Introduction
In algebra, the binomial theorem is a powerful tool for expanding expressions of the form (x+y)^n, where n is a positive integer. In this article, we will explore the binomial expansion of (x+1)^6.
Binomial Theorem
The binomial theorem states that:
$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$
where n is a positive integer, and k is an integer that ranges from 0 to n. The binomial coefficient binom{n}{k} is defined as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Expansion of (x+1)^6
Using the binomial theorem, we can expand (x+1)^6 as follows:
$(x+1)^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} 1^k$
Simplifying the expression, we get:
$(x+1)^6 = \binom{6}{0} x^6 + \binom{6}{1} x^5 + \binom{6}{2} x^4 + \binom{6}{3} x^3 + \binom{6}{4} x^2 + \binom{6}{5} x + \binom{6}{6}$
Evaluating the binomial coefficients, we get:
$(x+1)^6 = 1 x^6 + 6 x^5 + 15 x^4 + 20 x^3 + 15 x^2 + 6 x + 1$
Final Result
The binomial expansion of (x+1)^6 is:
$(x+1)^6 = x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1$
This expansion can be useful in various algebraic manipulations and applications.
