Find the general term in the expansion of (2x - 3/x)⁹

Question

Find the general term in the expansion of $\left(2x - \dfrac{3}{x}\right)^9$ .

(NCERT Class 11, Exercise 8.2)

Solution — Step by Step

For $(a + b)^n$ , the general term is:

T_{r+1} = \binom{n}{r} a^{n-r} b^r

Here $a = 2x$ , $b = -\dfrac{3}{x}$ , and $n = 9$ .

T_{r+1} = \binom{9}{r} (2x)^{9-r} \left(-\frac{3}{x}\right)^r

= \binom{9}{r} \cdot 2^{9-r} \cdot x^{9-r} \cdot (-3)^r \cdot x^{-r}

\boxed{T_{r+1} = \binom{9}{r} \cdot 2^{9-r} \cdot (-3)^r \cdot x^{9-2r}}

where $r = 0, 1, 2, \ldots, 9$ .

The power of $x$ is $9 - 2r$ . This tells us everything: the term independent of $x$ occurs when $9 - 2r = 0$ , i.e., $r = 4.5$ — which is not an integer, so there is no constant term in this expansion.

Why This Works

The binomial theorem systematically distributes the choice: in each of the 9 factors of $(2x - 3/x)$ , we pick either $2x$ or $-3/x$ . The $\binom{9}{r}$ counts the number of ways to pick $-3/x$ from exactly $r$ of the 9 factors (and $2x$ from the remaining $9-r$ ).

The power of $x$ in each term is $(9-r) - r = 9 - 2r$ because each $2x$ contributes $x^1$ and each $-3/x$ contributes $x^{-1}$ . This decreasing power pattern is useful for finding specific terms — the term with $x^3$ , the constant term, etc.

Alternative Method — Find a specific term

If the question asks for the coefficient of $x^3$ : set $9 - 2r = 3$ , so $r = 3$ .

T_4 = \binom{9}{3} \cdot 2^6 \cdot (-3)^3 \cdot x^3 = 84 \times 64 \times (-27) \cdot x^3 = -145152\,x^3

In JEE, binomial questions often ask for a specific term — “find the coefficient of $x^5$ ” or “find the term independent of $x$ .” Set the exponent of $x$ equal to the desired power and solve for $r$ . If $r$ isn’t a non-negative integer $\leq n$ , that term doesn’t exist.

Common Mistake

Students frequently drop the negative sign from $(-3/x)^r$ . The term $b = -3/x$ , not $3/x$ . When $r$ is odd, $(-3)^r$ is negative, and when $r$ is even, it’s positive. Forgetting this sign alternation will give wrong coefficients for odd- $r$ terms. Always write $(-3)^r$ explicitly and simplify the sign at the end.

Question

Solution — Step by Step

Why This Works

Alternative Method — Find a specific term

Common Mistake

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