Question

3)- (PUC-RS) o coeficiente de x² no desenvolvimento de (2x –1/x )6 é:
a) 16 b) 60 c) 160 d) 192 e) 240.

Answer 1

O termo geral de $\left( 2x-\dfrac{1}{x} \right)^6$ é $\dbinom{6}{k}\cdot(2x)^{6-k}\cdot\left( \dfrac{1}{x}\right)^{k}$.

Queremos $x^2$. Veja que $\dfrac{x^{6-k}}{x^k}=x^{6-k-k}=x^{6-2k}$
 
 Assim, devemos ter $x^{6-2k}=x^2 \iff 6-2k=2 \iff 2k=4 \iff k=2$.

O termo procurado é

$\dbinom{6}{2}\cdot(2x)^{6-2}\cdot\left( \dfrac{1}{x}\right)^{2}=\dfrac{6!}{2!\cdot4!}\cdot(2x)^4\cdot\left(\dfrac{1}{x}\right)^2=15\cdot16x^4\cdot\dfrac{1}{x^2} = 240x^2$

$\text{Alternativa E}$