1.
x²/(x⁴+x²+1)
=1/(x²+ 1+1/x²) /分子分母同除以x²
=1/[(x+ 1/x)²-1]
=1/(3²-1)
=1/8
2.
x+2y+4y²/(x-2y) -4x²y/(x²-4y²)
=[(x+2y)(x-2y)+4y²]/(x-2y)- 4x²y/[(x+2y)(x-2y)]
=(x²-4y²+4y²)/(x-2y) -4x²y/[(x+2y)(x-2y)]
=x²/(x-2y) -4x²y/[(x+2y)(x-2y)]
=[x²(x+2y)-4x²y]/[(x+2y)(x-2y)]
=x²(x+2y-4y)/[(x+2y)(x-2y)]
=x²(x-2y)/[(x+2y)(x-2y)]
=x²/(x+2y)
3.
1/[(x+1)(x+2)]+1/[(x+2)(x+3)]+1/[(x+3)(x+4)]+1/[(x+4)(x+5)]
=1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+1/(x+3)-1/(x+4)+1/(x+4)-1/(x+5)
=1/(x+1) -1/(x+5)
4.
規(guī)律：1/[n(n+1)]=1/n -1/(n+1) (n∈N+)
5.
1/[(x-2)(x-3)]-2/[(x-1)(x-3)]+1/[(x-1)(x-2)]
=1/(x-2)-1/(x-3) -[1/(x-1)-1/(x-3)]+1/(x-1)-1/(x-2)
=1/(x-2) -1/(x-3) -1/(x-1) +1/(x-3) +1/(x-1) -1/(x-2)
=0