Question

Determine o termo independente de x no desenvolvimento do binômio (x² + 1/x)9.

Answer 1

O termo independente é 84.

$\boxed{\text{Binômio\ de\ Newton} :( p+q)^{n} \ \sum _{k=0}^{n}\binom{n}{k} \cdotp p^{k} \cdotp q^{n-k}} \\ \\ \text{Temos} \ p=x^{2} ,q=\frac{1}{x} =x^{-1} \ \text{e} \ n=9\text{,\ logo} \ \left( x^{2} +x^{-1}\right)^{9} =\sum _{k=0}^{9}\binom{9}{k} \cdotp \left( x^{2}\right)^{k} \cdotp \left( x^{-1}\right)^{9-k} \\ \\ \text{Que\ simplificando\ fica} \ \sum _{k=0}^{9}\binom{9}{k} \cdotp x^{3k-9} \\ \\ \text{O\ termo\ independente\ é\ o\ coeficiente\ do\ termo\ em\ que\ o\ expoente\ de} \ x\text{\ é} \ 0 \\ \\ \text{\ Para} \ x^{3k-9} =x^{0}\text{,\ temos} \ 3k-9=0\Longrightarrow k=3 \\ \\ \text{E\ para} \ k=3\ \text{ficamos\ com} \ \binom{9}{3} \cdotp x^{0} =\binom{9}{3} =\frac{9!}{( 9-3) !\cdotp 3!} =\frac{9\cdotp 8\cdotp 7\cdotp \cancel{6!}}{\cancel{6!} \cdotp 3\cdotp 2\cdotp 1} =84 \\ \\ \therefore \boxed{\text{O\ termo\ independente\ do\ desenvolvimento\ de} \ \left( x^{2} +\frac{1}{x}\right)^{9} \ \text{é} \ 84}$