Question

Известно, что х+ 1/х=a. Найти х^3+1/х^3

Answer 1

$displaystyle tt x^3+frac{1}{x^3}=bigg(x+frac{1}{x}bigg)bigg(x^2-xcdotfrac{1}{x}+frac{1}{x^2}bigg)=bigg(x+frac{1}{x}bigg)bigg(x^2+frac{1}{x^2}-1bigg)=\ \ \ =bigg(x+frac{1}{x}bigg)bigg(x^2+2xcdotfrac{1}{x}+frac{1}{x^2}-2xcdotfrac{1}{x}-1bigg)=bigg(x+frac{1}{x}bigg)bigg(bigg(x+frac{1}{x}bigg)^2-3bigg)=\ \ \ =acdot (a^2-3)=a^3-3a$