分式的计算.求详细解答、解题思路及方法

题目

分式的计算.求详细解答、解题思路及方法
(1) x+2分之3 + 2-x分之1 +x平方-4分之2x
(2) (x-1分之x+1 - x+1分之x-1）÷ x平方-1分之x
(3) （a平方-2a分之a+2 - a平方-4a+4分之a-1）÷ a平方-2a分之4-a

答案

解(1)：
原式=3/(x+2)+1/(2-x)+2x/(x²-4) 通分,最简公分母是(x-2)(x+2)
=3(x-2)/[(x-2)(x+2)]-(x+2)/[(x-2)(x+2)]+2x/[(x-2)(x+2)]
=[3(x-2)-(x+2)+2x]/[(x-2)(x+2)]
=(3x-6-x-2+2x)/[(x-2)(x+2)]
=(4x-8)/[(x-2)(x+2)]
=4(x-2)/[(x-2)(x+2)] 约分
=4/(x+2)
解(2)：
原式=[(x+1)/(x-1)-(x-1)/(x+1)]÷[x/(x²-1)] 前面括号内通分
=[(x+1)²/(x²-1)-(x-1)²/(x²-1)]×[(x²-1)/x]
={[(x+1)²-(x-1)²]/(x²-1)}×[(x²-1)/x] 约分,合并
=(x²+2x+1-x²+2x-1)/x
=4x/x
=4
解(3)：
原式=[(a+2)/(a²-2a)-(a-1)/(a²-4a+4)]÷[(4-a)/(a²-2a)] 分解因式
={(a+2)/[a(a-2)]-(a-1)/(a-2)²}÷{(4-a)/[a(a-2)]} 通分
={(a+2)(a-2)/[a(a-2)²]-a(a-1)/[a(a-2)²]}×[a(a-2)/(4-a)]
={[(a+2)(a-2)-a(a-1)]/[a(a-2)²]}×[a(a-2)/(4-a)] 约分,合并
=[(a²-4-a²+a)/(a-2)]×[1/(4-a)]
=[(a-4)/(a-2)]×[-1/(a-4)] 约分
=-1/(a-2)