%5E5-binomial-expansion.jpeg "(x+1)^5 Binomial Expansion")
Binomial Expansion of (x+1)^5
In algebra, binomial expansion is a fundamental concept that deals with expanding powers of binomials, which are expressions consisting of two terms. One of the most popular binomial expansions is the expansion of (x+1)^5. In this article, we will explore how to expand (x+1)^5 using the binomial theorem.
What is the Binomial Theorem?
The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial. It states that for any positive integer n, the expansion of (x+y)^n is given by:
$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$
where n is a positive integer, x and y are variables, and \binom{n}{k} is the binomial coefficient.
Expanding (x+1)^5
To expand (x+1)^5, we can use the binomial theorem with n=5, x=x, and y=1. Substituting these values into the binomial theorem formula, we get:
$(x+1)^5 = \sum_{k=0}^{5} \binom{5}{k} x^{5-k} 1^k$
Simplifying the expression, we obtain:
$(x+1)^5 = \binom{5}{0} x^5 + \binom{5}{1} x^4 + \binom{5}{2} x^3 + \binom{5}{3} x^2 + \binom{5}{4} x + \binom{5}{5}$
Calculating the Binomial Coefficients
To calculate the binomial coefficients, we can use the formula:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Substituting n=5 and k=0, 1, 2, 3, 4, 5, we get:
$\binom{5}{0} = 1$ $\binom{5}{1} = 5$ $\binom{5}{2} = 10$ $\binom{5}{3} = 10$ $\binom{5}{4} = 5$ $\binom{5}{5} = 1$
Final Expansion
Substituting the binomial coefficients into the expansion, we finally get:
$(x+1)^5 = x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1$
And that's it! We have successfully expanded (x+1)^5 using the binomial theorem.
Conclusion
Expanding (x+1)^5 is a straightforward process using the binomial theorem. By applying the formula and calculating the binomial coefficients, we can obtain the final expansion. This expansion has numerous applications in algebra, calculus, and other areas of mathematics.
