Question

(Ufam) O termo independente de x, no desenvolvimento do binômio [(1/x + x)(1/x - x) é igual a:
a)-70          d)20
b)70           e)60
c)-20

Answer 1

$[(\frac{1}{x}+x).(\frac{1}{x}-x)]^8\\\\\ (x^-^2-x^2)^8\\\\\ T_p_+_1 = (^8_p)x^-^2(8-p)x^2(p)\\\\\ T_p_+_1 = (^8_p)x^(^-^1^6^+^2^p)x^2^p\\\\\ T_p_+_1 = (^8_p)x^(^-^1^6^+^4^p^)$

$x^(^-^1^6^+^4^p^) = x^0\\\\\ -16+4p = 0\\\ 4p = 16\\\\ \boxed{p=4}$

Substituindo:

$T_5 = (^8_4)x^(^-^1^6^+^1^6^)\\\\\\ T_5 = (^8_4)x^0\\\\\\ (^8_4) = \frac{8!}{4!4!} = 70\\\\\ \boxed{T_5 = 70}$