\(x+\dfrac{1}{x}=a\Rightarrow\left(x+\dfrac{1}{x}\right)^3=a^3\)
\(\Rightarrow x^3+\dfrac{1}{x^3}+3x\dfrac{1}{x}\left(x+\dfrac{1}{x}\right)=a^3\)
\(\Leftrightarrow x^3+\dfrac{1}{x^3}=a^3-3a\)
\(x^2+\dfrac{1}{x^2}=\left(x+\dfrac{1}{x}\right)^2-2x\dfrac{1}{x}=a^2-2\)
\(x^4+\dfrac{1}{x^4}=\left(x^2\right)^2+\left(\dfrac{1}{x^2}\right)^2=\left(x^2+\dfrac{1}{x^2}\right)^2-2x^2\dfrac{1}{x^2}=\left(a^2-2\right)^2-2\)\(\Rightarrow\left(x^3+\dfrac{1}{x^3}\right)\left(x^4+\dfrac{1}{x^4}\right)=\left(a^3-3a\right)\left(a^2-2\right)^2\)\(\Leftrightarrow x^7+\dfrac{x^3}{x^4}+\dfrac{x^4}{x^3}+\dfrac{1}{x^7}=\left(a^3-3a\right)\left(a^2-2\right)^2\)\(\Rightarrow x^7+\dfrac{1}{x^7}=\left(a^3-3a\right)\left(a^2-2\right)^2-a\)
Thuộc kết quả mấy cái bài như thế này mất:(
