Expanding the Binomial Expression: (2x - 1/x10) and Finding the 4th Term

When dealing with polynomial expressions and their expansions, the use of the binomial theorem is a powerful tool. This article will guide you through the process of expanding the expression (2x - 1/x10) and determining the 4th term using the binomial theorem.

Understanding the Binomial Theorem

The binomial theorem is a formula that describes the algebraic expansion of powers of a binomial, which is an expression consisting of two terms. The general form of the theorem is given by:

((a b)^n sum_{k0}^{n} binom{n}{k} a^{n-k} b^k)

Applying the Binomial Theorem to (2x - 1/x10)

In our case, the expression is:

((2x - frac{1}{x})^{10})

Here, we have:

(a 2x) (b -frac{1}{x}) (n 10)

The nth term of the expansion is given by:

(T_{k 1} binom{n}{k} a^{n-k} b^k)

To find the 4th term, we set (k 3).

Calculating the 4th Term

The 4th term is:

(T_4 binom{10}{3} (2x)^{10-3} left(-frac{1}{x}right)^3)

Breaking this down step by step:

(binom{10}{3} frac{10!}{3!(10-3)!} frac{10 cdot 9 cdot 8}{3 cdot 2 cdot 1} 120) ((2x)^7 2^7 x^7 128x^7) (left(-frac{1}{x}right)^3 -frac{1}{x^3})

Putting it all together:

(T_4 120 cdot 128x^7 cdot -frac{1}{x^3} -120 cdot 128 cdot x^{7-3} -1536^4)

Final Answer

The 4th term of the expression (2x - 1/x10) is:

(boxed{-1536^4})

Conclusion

Using the binomial theorem correctly and carefully calculating each part, we have determined the 4th term of the given expression. This process can be applied to other similar problems in algebra and polynomial expansions.