Đề bài
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
Lời giải chi tiết
a) 23x+xx−1+6x2−42x(1−x)
=23x+xx−1+3x2−2x(1−x)
=23x+xx−1−3x2−2x(x−1)
=2(x−1)+x.3x−3(3x2−2)3x(x−1)
=2x−2+3x2−9x2+63x(x−1)
=−6x2+2x+43x(x−1)
=−2(3x2−x−2)3x(x−1)
=−2(3x2−3x+2x−2)3x(x−1)
=−2[3x(x−1)+2(x−1)]3x(x−1)
=−2(3x+2)(x−1)3x(x−1)
=−2(3x+2)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
