Find the term that does not contain x in the expansion of $$ \left(4x-\frac{1}{2x}\right)^{10} $$


To find the term that does not contain $ x $ in the expansion of $ \left(4x - \frac{1}{2x}\right)^{10} $, we'll use the Binomial Theorem. The Binomial Theorem states that:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

Here, $ a = 4x $, $ b = -\frac{1}{2x} $, and $ n = 10 $. 

The general term in the expansion is given by:

\[
T_k = \binom{10}{k} (4x)^{10-k} \left(-\frac{1}{2x}\right)^k
\]

Simplifying the general term:

\[
T_k = \binom{10}{k} (4^{10-k} x^{10-k}) \left(-\frac{1}{2^k x^k}\right)
\]

\[
T_k = \binom{10}{k} (-1)^k \frac{4^{10-k}}{2^k} x^{10-k-k}
\]

\[
T_k = \binom{10}{k} (-1)^k \frac{4^{10-k}}{2^k} x^{10-2k}
\]

We need the term that does not contain $ x $, which means the exponent of $ x $ must be zero:

\[
10 - 2k = 0
\]

Solving for $ k $:

\[
2k = 10 \implies k = 5
\]

So, the term that does not contain $ x $ occurs when $ k = 5 $. Substituting $ k = 5 $ into the expression for $ T_k $:

\[
T_5 = \binom{10}{5} (-1)^5 \frac{4^{5}}{2^5} x^{0}
\]

Simplifying further:

\[
T_5 = \binom{10}{5} (-1)^5 \frac{1024}{32}
\]

\[
T_5 = \binom{10}{5} (-1)^5 \cdot 32
\]

\[
T_5 = 252 \times (-32) = -8064
\]

Thus, the term that does not contain $ x $ in the expansion is $-8064$.

Would you like more details on this or have any questions?

Here are 8 related questions:
1. How do you expand a binomial expression using the Binomial Theorem?
2. What is the general term in the binomial expansion?
3. How can you determine the coefficient of a specific term in the expansion?
4. What is the significance of the binomial coefficient $\binom{n}{k}$?
5. How do you handle negative exponents in binomial expansions?
6. Can the Binomial Theorem be used for non-integer exponents?
7. How do you find the constant term in a binomial expansion?
8. What is the role of the sign in the binomial expansion?

**Tip:** In binomial expansions, identifying the term that does not contain the variable often involves setting the exponent of the variable to zero.

Explore how to find the term that does not contain x in the expansion of (4x - 1/(2x))^10 using the Binomial Theorem. Step-by-step solution provided, including simplification and calculation of the specific term. Suitable for advanced high school students and beyond.

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