Question

Qual o termo independente de
x

Answer 1

O termo geral é dado por:

$\text{T}_{k+1}=\dbinom{n}{k}a^{n-k}\cdot b^{k}$

Nessa questão temos $a=x^3$, $b=\dfrac{1}{x^2}=x^{-2}$ e $n=10$

Substituindo:

$\text{T}_{k+1}=\dbinom{10}{k}(x^3)^{10-k}\cdot (x^{-2})^{k}=\dbinom{10}{k}x^{30-3k}\cdot x^{-2k}$

Assim, devemos ter:

$x^{30-3k}\cdot x^{-2k}=x^{0}~\longrightarrow~x^{30-3k-2k}=x^{0}$

$x^{30-5k}=x^{0}~\longrightarrow~30-5k=0$

$5k=30~\longrightarrow~k=\dfrac{30}{5}~\longrightarrow~k=6$

O termo independente é $\dbinom{10}{6}(x^4)^{3}\cdot (x^{-2})^{6}=210\cdot x^{12}\cdot x^{-12}=210$