Find the term independent of x in the expansion (x-1/x^2)^9

To find the term independent of \( x \) in the expansion of \( \left(x - \frac{1}{x^2}\right)^9 \), we will use the Binomial Theorem. The Binomial Theorem states:

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

For the given expression, \( a = x \) and \( b = -\frac{1}{x^2} \), and \( n = 9 \).

The general term in the expansion is:

\[
T_k = \binom{9}{k} x^{9-k} \left(-\frac{1}{x^2}\right)^k
\]

Simplifying this:

\[
T_k = \binom{9}{k} x^{9-k} \cdot (-1)^k \cdot x^{-2k}
\]

\[
T_k = \binom{9}{k} (-1)^k x^{9-k-2k} = \binom{9}{k} (-1)^k x^{9-3k}
\]

We want the term that is independent of \( x \), which means the exponent of \( x \) should be 0:

\[
9 - 3k = 0
\]

Solving for \( k \):

\[
3k = 9 \implies k = 3
\]

Now, substitute \( k = 3 \) back into the general term:

\[
T_3 = \binom{9}{3} (-1)^3 x^{9-3(3)} = \binom{9}{3} (-1)^3 x^0
\]

\[
T_3 = \binom{9}{3} (-1)^3 = \binom{9}{3} (-1) = 84(-1) = -84
\]

Therefore, the term independent of \( x \) is \(\boxed{-84}\).

Would you like any further details or have questions about this solution?

Here are 5 related questions you might find interesting:
1. How would the expansion change if the power were different, say \( (x - \frac{1}{x^2})^{10} \)?
2. What would the coefficient of \( x^3 \) be in the expansion of \( \left(x - \frac{1}{x^2}\right)^9 \)?
3. Can you find the constant term in the expansion of \( \left(2x - \frac{3}{x^2}\right)^9 \)?
4. How would you approach finding the middle term in the expansion of \( \left(x - \frac{1}{x^2}\right)^9 \)?
5. What is the sum of all coefficients in the expansion of \( \left(x - \frac{1}{x^2}\right)^9 \)?

**Tip:** When using the Binomial Theorem, identifying the general term can be crucial in solving many types of binomial expansion problems efficiently.

Find the term independent of x in the expansion (x - 1/x^2)^9.

Explore how to find the term independent of x in the binomial expansion of (x - 1/x^2)^9. This detailed solution covers the application of the Binomial Theorem and exponent manipulation, suitable for high school students (Grades 10-12).

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