Question

5- O termo independente de x no desenvolvimento do binômio (x/2 / 2/x) elevado a 12

a) 4096
b) 2048
c) 1024
d) 1136
e) 924

Answer 1

Explicação passo-a-passo:

O termo geral é:

$\sf T_{p+1}=\dbinom{12}{p}\cdot\Big(\dfrac{x}{2}\Big)^{12-p}\cdot\Big(\dfrac{2}{x}\Big)^{p}$

Temos que:

$\sf x^{12-p}\cdot\dfrac{1}{x^{p}}=x^0$

$\sf \dfrac{x^{12-p}}{x^{p}}=x^0$

$\sf x^{12-p-p}=x^0$

$\sf x^{12-2p}=x^0$

Igualando os expoentes:

$\sf 12-2p=0$

$\sf 2p=12$

$\sf p=\dfrac{12}{2}$

$\sf p=6$

O termo independente é:

$\sf T_{p+1}=\dbinom{12}{p}\cdot\Big(\dfrac{x}{2}\Big)^{12-p}\cdot\Big(\dfrac{2}{x}\Big)^{p}$

$\sf T_{6+1}=\dbinom{12}{6}\cdot\Big(\dfrac{x}{2}\Big)^{12-6}\cdot\Big(\dfrac{2}{x}\Big)^{6}$

$\sf T_{7}=924\cdot\Big(\dfrac{x}{2}\Big)^{6}\cdot\Big(\dfrac{2}{x}\Big)^{6}$

$\sf T_{7}=924\cdot\dfrac{x^6}{2^6}\cdot\dfrac{2^6}{x^6}$

$\sf \red{T_{7}=924}$

Letra E