Question

Determine o coeficiente do termo em x8, no desenvolvimento de $( x^{2} +1/x)^1^0.$

Answer 1

$\large\begin{array}{l} \textsf{Temos o seguinte bin\^omio:}\end{array}$

$\mathsf{\left(x^2+\dfrac{1}{x}\right)^{10}}\qquad \quad\mathsf{a=x^2}\qquad \mathsf{b=\dfrac{1}{x}}\qquad \mathsf{n=10}$


$\large\begin{array}{l} \textsf{Um termo qualquer obtido no desenvolvimento \'e dado por:}\end{array}$

$\mathsf{T_{p+1}=C_{n,p}\cdot a^{n-p}\cdot b^p}$


$\large\begin{array}{l} \textsf{Para o dado bin\^omio, temos:}\end{array}$

$\mathsf{T_{p+1}=C_{10,p}\cdot {(x^2)}^{10-p}\cdot \left(\dfrac{1}{x}\right)^p}\\\\ \mathsf{T_{p+1}=C_{10,p}\cdot x^{2\cdot (10-p)}\cdot x^{-p}}\\\\ \mathsf{T_{p+1}=C_{10,p}\cdot x^{2\cdot (10-p)-p}\qquad(i)}$


$\large\begin{array}{l} \textsf{Veja que o termo que buscamos tem }\mathbf{x^8}\textsf{, logo:}\end{array}$

$\mathsf{x^{2\cdot (10-p)-p}=x^8}}$


$\large\begin{array}{l} \textsf{Resolvendo a equa\c{c}\~ao exponencial:}\end{array}$

$\mathsf{2\cdot (10-p)-p=8}\\\\ \mathsf{20-2p-p=8}\\\\ \mathsf{-3p=8-20}\\\\ \mathsf{-3p=-12}\\\\ \mathsf{p=4}$


$\large\begin{array}{l} \textsf{Substituindo o valor de }\mathbf{p}\textsf{ em }\mathbf{(i)}\textsf{, vem:} \end{array}$

$\mathsf{T_{p+1}=C_{10,p}\cdot x^{2\cdot (10-p)-p}}\\\\ \mathsf{T_{4+1}=C_{10,4}\cdot x^{8}}\end{array}$

$\mathsf{T_{5}=\dfrac{10!}{4!\cdot 6!}\cdot x^{8}}\\\\\\ \mathsf{T_{5}=\dfrac{10\cdot 9\cdot 8\cdot 7 \cdot \diagup\!\!\!\!6!}{4\cdot 3\cdot 2\cdot \diagup\!\!\!\!6!}\cdot x^{8}}\\\\\\ \mathsf{T_{5}=\dfrac{10\cdot \diagdown\!\!\!\!\!9^3\cdot \diagup\!\!\!\!8\cdot 7}{\diagup\!\!\!\!4\cdot \diagdown\!\!\!\!3\cdot \diagup\!\!\!\!2}\cdot x^{8}}\\\\\\ \mathsf{T_5=210x^8}$


$\large\begin{array}{l} \textsf{Resposta: o coeficiente \'e 210} \end{array}$


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$\large\begin{array}{l} \textsf{Bons estudos! :)} \end{array}$


Tags: binômio de newton encontrar termo coeficiente