Rút gọn biểu thức sau:
a) 23x+xx−1+6x2−42x(1−x)
b) x3+11−x3+xx−1−x+1x2+x+1
c) (2x+2−21−x).x2−44x2−1
d) 1+x3−xx2+1(11−x−11−x2)
Thực hiện theo quy tắc cộng, trừ, nhân, chia các phân thức đại số
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a) 23x+xx−1+6x2−42x(1−x) =23x+−x1−x+3x2−2x(1−x) =2−2x−3x2+9x2−63x(1−x) =6x2−2x−43x(1−x) =2(3x+1)3x
b) x3+11−x3+xx−1−x+1x2+x+1 =−x3−1x3−1+xx−1−x+1x2+x+1 =−x3−1+x(x2+x+1)−(x2−1)(x−1)(x2+x+1) =−x3−1+x3+x2+x−x2+1(x−1)(x2+x+1) =xx3−1
c) Ta có:
2x+2−21−x =2(1−x)−2(x+2)(x+2)(1−x) =2−2x−2x−4(x+2)(1−x) =−4x−2(x+2)(1−x) =2(2x+1)(x+2)(x−1);
x2−44x2−1 =(x−2)(x+2)(2x−1)(2x+1).
Do đó (2x+2−21−x).x2−44x2−1 =2(2x+1)(x+2)(x−1).(x−2)(x+2)(2x−1)(2x+1) =2(x−2)(2x−1)(x−1)
d) 1+x3−xx2+1(11−x−11−x2) =1+x3−xx2+1(11−x−11−x2) =1+x3−xx2+1.1+x−11−x2 =1+x(x2−1)x2+1.x1−x2 =1+−x2(x2−1)(x2+1)(x2−1) =1+−x2x2+1 =x2+1−x2x2+1 =1x2+1