Question

URGENTE! POR FAVOR, AJUDA AQUI.

Sendo ( $x + \frac{1}{x} )^{2} = 3, calcule x^{3} + \frac{1}{x^{3} }.$

EXPLICAÇÃO!!!

Answer 1

$\displaystyle (\text x+\frac{1}{\text x})^2 = 3 \ \ ; \ \ \text x^3+\frac{1}{\text x^3}=\ ? \\\\ \underline{\text{desenvolvendo}}: \\\\ (\text x +\frac{1}{\text x})^2 = \text x^2 + 2.\text x.\frac{1}{\text x} +\frac{1}{\text x^2} \\\\\\ 3 = \text x^2 + 2+\frac{1}{\text x^2} \\\\\\ \text x^2+\frac{1}{\text x^2} = 3-2 \\\\\\ \text x^2+\frac{1}{\text x^2} = 1$

Usando soma de cubos :

$\text a^3+\text b^3=(\text a+\text b)(\text a^2-\text{a.b}+\text b^2)$

Fazendo a = x e b = 1/x :

$\displaystyle \text x^3+\frac{1}{\text x^3} =(\text x+\frac{1}{\text x})(\text x^2-\text x.\frac{1}{\text x}+\frac{1}{\text x^2}) \\\\\\ \text x^3+\frac{1}{\text x^3} = (\text x+\frac{1}{\text x})(\text x^2-1+\frac{1}{\text x^2}) \\\\\ \underline{\text{temos}}: \\\\ (\text x+\frac{1}{\text x})^2 = 3 \to (\text x+\frac{1}{\text x}) = \pm\sqrt 3\\\\ \therefore \\\\\ \text x^3+\frac{1}{\text x^3} = \pm\sqrt3(1-1) \\\\\\ \text x^3 + \frac{1}{\text x^3} = \pm\sqrt{3}.0$

Portanto :

$\huge\boxed{\ \text x^3+\frac{1}{\text x^3}=0 \ }\checkmark$