Binomial Theorem: Edge Cases and Subtle Traps (5)

Question

Find the term independent of $x$ in the expansion of $\left(x^2 - \frac{1}{x}\right)^9$.

Solution — Step by Step

Why This Works

The general term packages all the binomial structure in one expression. To find a specific feature (constant term, coefficient of $x^k$, middle term), we just set up an equation in $r$ and solve.

The $(-1)^r$ factor matters when $r$ is odd — many students forget this and report a wrong sign. Here $r = 6$ is even, so the sign stays positive.

Alternative Method

Multinomial expansion or generating functions — both work but are massive overkill for a single-term query. The general term approach is the standard JEE technique.

Common Mistake

The classic error: forgetting to handle the sign in $(-1/x)^r$. The negative sign carries an extra factor of $(-1)^r$. If the question asked for the coefficient of $x^3$, we’d find $r = 5$ and the sign would be $(-1)^5 = -1$ — a different answer than the unsigned binomial coefficient.