f(x+(1/x))= x^3+(1/x^3)= (x+(1/x))^3-3(x+(1/x)) 怎么化成这步的 --------------------------
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/06 12:24:00
![f(x+(1/x))= x^3+(1/x^3)= (x+(1/x))^3-3(x+(1/x)) 怎么化成这步的 --------------------------](/uploads/image/z/6902604-36-4.jpg?t=f%28x%2B%281%2Fx%29%29%3D+x%5E3%2B%281%2Fx%5E3%29%3D+%28x%2B%281%2Fx%29%29%5E3-3%28x%2B%281%2Fx%29%29+%E6%80%8E%E4%B9%88%E5%8C%96%E6%88%90%E8%BF%99%E6%AD%A5%E7%9A%84+--------------------------)
f(x+(1/x))= x^3+(1/x^3)= (x+(1/x))^3-3(x+(1/x)) 怎么化成这步的 --------------------------
f(x+(1/x))= x^3+(1/x^3)= (x+(1/x))^3-3(x+(1/x)) 怎么化成这步的
--------------------------
f(x+(1/x))= x^3+(1/x^3)= (x+(1/x))^3-3(x+(1/x)) 怎么化成这步的 --------------------------
x^3+(1/x^3)
= x^3 + (1/x)^3 + 3(x^2)*(1/x) + 3x*(1/x)^2 - 3(x^2)*(1/x) - 3x*(1/x)^2
= ( x^3 + (1/x)^3 + 3(x^2)*(1/x) + 3x*(1/x)^2 ) - 3x - 3/x
= (x + 1/x)^3 - 3(x + 1/x)
x^3+1/x^3
=(x+1/x)(x^2-x*1/x+1/x^2)
=(x+1/x)[(x^2+1/x^2+2x*1/x)-3]
=(x+1/x)[(x+1/x)^2-3]
=(x+1/x)^3-3(x+1/x)
所用公式:a^3+b^3=(a^2-ab+b^2)
x^3+1/x^3
立方和
=(x+1/x)(x^2-1+1/x^2)
=(x+1/x)[(x+1/x)^2-2-1]
=(x+1/x)^3-3(x+1/x)
用立方和公式。简单。稍等。
这个其实是公式(a+b)^3 = a^3 +b^3 + 3(a^2)b+3a(b^2) 的逆运用
x^3+(1/x^3)=(x+1/x)(x^2-1+1/x^2)=(x+1/x)(x^2+2+1/x^2-3)=(x+1/x)[(x+1/x)^2-3]=(x+1/x)^3-3(x+1/x)
第一步是根据立方和公式。