Question

determinado o binômio de newton para o 4termo no desenvolvimento (3r^2 - 4s^3 )^5

Answer 1

Olá Grasielly, bom dia!

Da definição de Binômio de Newton, temos:

$\\ (3r^2 - 4s^3)^5 = \binom{5}{0} \cdot (3r^2)^{5 - 0} \cdot (4s^3)^0 + \binom{5}{1} \cdot (3r^2)^{5 - 1} \cdot (4s^3)^1 + \binom{5}{2} \cdot (3r^2)^{5 - 2} \cdot (4s^3)^2 + \binom{5}{3} \cdot (3r^2)^{5 - 3} \cdot (4s^3)^3 + \binom{5}{4} \cdot (3r^2)^{5 - 4} \cdot (4s^3)^4 + \binom{5}{5} \cdot (3r^2)^{5 - 5} \cdot (4s^3)^5$

 Portanto,

$\\ \binom{5}{4} \cdot (3r^2)^{5 - 4} \cdot (4s^3)^4 = \\\\\\ \frac{5!}{(5 - 4)!4!} \cdot (3r^2)^1 \cdot (4s^3)^4 = \\\\\\ \frac{5 \cdot 4!}{1!4!} \cdot 3r^2 \cdot (4s^3)^4 = \\\\\\ (5 \cdot 3 \cdot 256) \cdot r^2 \cdot s^{12} = \\\\\\ \boxed{3840r^2s^{12}}$