Binomial Theorem: Tricky Questions Solved (1)

Question

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

Solution — Step by Step

The term independent of $x$ is $672$.

Why This Works

In any binomial expansion $(A + B)^n$, the general term is $T_{r+1} = \binom{n}{r}A^{n-r}B^r$. Whenever a problem asks for the term with a specific power of $x$ (or independent of $x$), set the exponent of $x$ in $T_{r+1}$ equal to the required value and solve for $r$.

If $r$ comes out as an integer in $\{0, 1, \ldots, n\}$, that term exists. If $r$ is not an integer or out of range, no such term exists.

Alternative Method

Brute force expansion of $(2x^2 - 1/x)^9$ is impractical with $10$ terms each having complicated coefficients. The general-term approach is the only sensible path.

Common Mistake

Forgetting the $(-1)^r$ from the negative sign in $-1/x$. Students compute the magnitude correctly but get the sign wrong. The fix: write all factors explicitly, including signs, before simplifying.