Question

Determine o termo independente de x no desenvolviento de $\left\ (x^{2} + \frac{1}{x} ) \right\) ^{6}$

Answer 1

resolvendo:

T k+1 = |6|(x^2)^6-k (1/x)^k
----------- |k| --


=> T k+1 = |6| (x)^12-2k / (x)^k =>
-----------------|k| --

=> T k+1 = |6| x^12-3k
-----------------|k|--

para que T k+1 seja independente de X, devemos ter:
12 - 3k = 0
- 3k = - 12 .(-1)
3k = 12
k = 12/3
k = 4

Logo, o termo independente é:
T k+1 = |6| x^0
------------|4|--


T 5 = |6| x^0 = 6! /4!(6-4)!
--------|4|--

= 6! / 4!2! = 6*5*4*3*2! / 4*3*2! 2!
= 6*5 / 2 => 30/2 => 15

logo:
T 5 = |6| x^0
---------|4|--

T 5 = 15