怎么用凑配法解此题?已知f(x+1/x)=x^3+1/x^3+2,求f(x)

怎么用凑配法解此题?已知f(x+1/x)=x^3+1/x^3+2,求f(x)
怎么用凑配法解此题?
已知f(x+1/x)=x^3+1/x^3+2,求f(x)

怎么用凑配法解此题?已知f(x+1/x)=x^3+1/x^3+2,求f(x)
令a=x+1/x
则a²=x²+2+1/x²
x²+1/x²=a²-2
所以x³+1/x³
=(x+1/x)(x²-1+1/x²)
=a(a²-2-1)
所以f(a)=a(a²-3)+2=a³-3a+2
所以f(x)=x³-3x+2

f(x+1/x)=x^3+1/x^3+2
=(x+1/x)(x^2-1+1/x^2)+2
=(x+1/x)[(x+1/x)^2-3]+2
故：f(x)=x(x^2-3)+2
=x^3-3x+2