Calcule utilizando a fórmula do binômio de newton: [4- ] ³.... Pergunta de ideia deZene17

January 2020 1 100 Report

Calcule utilizando a fórmula do binômio de newton:

[4- $(2)^{ \frac{1}{2} }$ ] ³

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O desenvolvimento do Binômio de Newton

$\mathsf{(v+w)^n}$

possui n + 1 termos, e o termo da posição p + 1 é dado por

$\mathsf{t_{p+1}=\dbinom{n}{p}\cdot v^{n-p}\cdot w^p}$

A expansão é dada por

$\mathsf{(v+w)^n=\displaystyle\sum_{p=0}^n\binom{n}{p}\cdot v^{n-p}\cdot w^p}\\\\\\\mathsf{(v+w)^n=\displaystyle\sum_{p=0}^n t_{p+1}}$

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Expandir

$\mathsf{(4-2^{1/2})^3\quad\longrightarrow\quad v=4,~~w=-2^{1/2},~~n=3}$

O desenvolvimento terá 3 + 1 = 4 termos.

• 1º termo:

$\mathsf{t_1=\dbinom{3}{0}\cdot 4^3\cdot (-2^{1/2})^0}\\\\\\ \mathsf{t_1=1\cdot 64\cdot 1}\\\\ \mathsf{t_1=64}$

• 2º termo:

$\mathsf{t_2=\dbinom{3}{1}\cdot 4^{3-1}\cdot (-2^{1/2})^1}\\\\\\ \mathsf{t_2=3\cdot 4^2\cdot (-2^{1/2})}\\\\ \mathsf{t_2=3\cdot 16\cdot (-2^{1/2})}\\\\\mathsf{t_2=-\,48\cdot 2^{1/2}}$

• 3º termo:

$\mathsf{t_3=\dbinom{3}{2}\cdot 4^{3-2}\cdot (-2^{1/2})^2}\\\\\\ \mathsf{t_3=3\cdot 4^1\cdot (-2^{1/2})^2}\\\\ \mathsf{t_3=3\cdot 4\cdot 2^1}\\\\\mathsf{t_3=24}$

• 4º termo:

$\mathsf{t_4=\dbinom{3}{3}\cdot 4^{3-3}\cdot (-2^{1/2})^3}\\\\\\ \mathsf{t_4=1\cdot 4^0\cdot (-2^{1/2})^3}\\\\ \mathsf{t_4=1\cdot 1\cdot (-2^{3/2})}\\\\\mathsf{t_4=-\,2^{3/2}}$

Portanto,

$\mathsf{(4-2^{1/2})^3=64-48\cdot 2^{1/2}+24-2^{3/2}}\\\\ \mathsf{(4-2^{1/2})^3=64+24-48\cdot 2^{1/2}-2^{1+(1/2)}}\\\\ \mathsf{(4-2^{1/2})^3=88-48\cdot 2^{1/2}-2\cdot 2^{1/2}}\\\\ \mathsf{(4-2^{1/2})^3=88-(48+2)\cdot 2^{1/2}}$

$\mathsf{(4-2^{1/2})^3=88-50\cdot 2^{1/2}}\quad\longleftarrow\quad\textsf{esta \'e a resposta.}$

Bons estudos! :-)