Question

x^5+1/x^5 怎么化解成（X+1/X）^5-5(x^3+1/x^3) -10(X+1/X) 求数学大神 帮帮忙

谢谢 大神们

Answer 1

(a+b)^2=a^2+2ab+b^2
(a+b)^3=a^3+3a^2b+3ab^2+b^3
(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4
(a+b)^5=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5……这是杨辉三角
所以(x+1/x)^5
=x^5+5x^4(1/x)+10x^3(1/x)^2+10x^2(1/x)^3+5x(1/x)^4+(1/x)^5
=x^5+5x^3+10x+10(1/x)+5(1/x)^3+(1/x)^5
=(x^5+1/x^5)+5(x^3+1/x^3)+10(x+1/x)
所以x^5+1/x^5 =（X+1/X）^5-5(x^3+1/x^3) -10(X+1/X)