Question

Wiadomo że [tex]x+ \frac{1}{x} = 4[/tex] . Oblicz x^3 + 1/x^3

Answer 1

[tex]\left(x+\dfrac{1}{x}\right)^3=x^3+3x+\dfrac{3}{x}+\dfrac{1}{x^3}=x^3+\dfrac{1}{x^3}+3\left(x+\dfrac{1}{x}\right)[/tex]

Zatem

[tex]x^3+\dfrac{1}{x^3}=\left(x+\dfrac{1}{x}\right)^3-3\left(x+\dfrac{1}{x}\right)[/tex]

A więc

[tex]x^3+\dfrac{1}{x^3}=4^3-3\cdot4=64-12=52[/tex]