The Importance of the Coefficient of 1/x in the Expansion of 2x - 1/x^5

Understanding the coefficient of 1/x in the expansion of 2x - 1/x^5 is essential in various mathematical contexts, including algebra and calculus. This article explores the detailed steps to find the coefficient, and explains its significance in mathematical problems and applications.

Introduction to the Binomial Theorem

The binomial theorem is a valuable tool for expanding expressions of the form (a b)^n. The theorem states that:

[Formula: (a b)^n sum_{k0}^{n} binom{n}{k} a^{n-k} b^k]

Where (binom{n}{k}) represents the binomial coefficient, which can be calculated using the formula:

(binom{n}{k} frac{n!}{k!(n-k)!})

This article will use the binomial theorem to find the coefficient of 1/x in the expansion of 2x - 1/x^5.

Step-by-Step Process to Find the Coefficient

Given the expression 2x - 1/x^5, we can expand it using the binomial theorem. Let's assign the terms in the expression as follows:

The binomial expansion will be:

[Formula: (2x (-1/x))^5 sum_{k0}^{5} binom{5}{k} (2x)^{5-k} left(-frac{1}{x}right)^k]

We now simplify the terms in the sum:

begin{aligned} (2x (-1/x))^5 sum_{k0}^{5} binom{5}{k} (2x)^{5-k} left(-frac{1}{x}right)^k sum_{k0}^{5} binom{5}{k} 2^{5-k} x^{5-k} cdot (-1)^k cdot frac{1}{x^k} sum_{k0}^{5} binom{5}{k} (-1)^k 2^{5-k} x^{5-k-k} sum_{k0}^{5} binom{5}{k} (-1)^k 2^{5-k} x^{5-2k} end{aligned} end{equation}

We are interested in the coefficient of 1/x, which occurs when 5 - 2k -1. Solving for k, we get:

begin{aligned} 5 - 2k -1 6 2k k 3 end{aligned} end{equation}

Now, we substitute k 3 into the binomial expansion to find the coefficient:

begin{aligned} text{Coefficient} binom{5}{3} (-1)^3 2^{5-3} 10 cdot (-1) cdot 4 -40 end{aligned} end{equation}

Thus, the value of the coefficient of 1/x in the expansion of 2x - 1/x^5 is -40.

Conclusion

The coefficient of 1/x in the expansion of 2x - 1/x^5 is -40. This result highlights the importance of understanding the binomial theorem and its applications. Understanding such coefficients can help in solving complex equations and in various fields of mathematics and physics.