%5E5-binomial-expansion.jpeg "(x/3+1/x)^5 Binomial Expansion")
Binomial Expansion of (x/3+1/x)^5
Introduction
In algebra, binomial expansion is a fundamental concept used to expand powers of a binomial expression. In this article, we will explore the binomial expansion of (x/3+1/x)^5.
The Binomial Theorem
The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial expression. It is stated as follows:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where $a$ and $b$ are real numbers, $n$ is a positive integer, and $\binom{n}{k}$ is the binomial coefficient.
Expanding (x/3+1/x)^5
Using the binomial theorem, we can expand (x/3+1/x)^5 as follows:
$(x/3+1/x)^5 = \sum_{k=0}^5 \binom{5}{k} (x/3)^{5-k} (1/x)^k$
Simplifying the expression, we get:
$(x/3+1/x)^5 = \binom{5}{0} (x/3)^5 + \binom{5}{1} (x/3)^4 (1/x) + \binom{5}{2} (x/3)^3 (1/x)^2 + \binom{5}{3} (x/3)^2 (1/x)^3 + \binom{5}{4} (x/3) (1/x)^4 + \binom{5}{5} (1/x)^5$
Simplifying the Terms
Simplifying each term, we get:
$= \frac{x^5}{243} + 5\frac{x^4}{81} + 10\frac{x^3}{27} + 10\frac{x^2}{9} + 5\frac{x}{3} + 1$
Conclusion
In this article, we have successfully expanded (x/3+1/x)^5 using the binomial theorem. The result is a polynomial expression with terms in increasing powers of x. Understanding binomial expansion is crucial in algebra and has numerous applications in mathematics, physics, and engineering.
