Giải thích các bước giải:

c.Ta có:

$\dfrac{x-1}{x^3}-\dfrac{x+1}{x^3-x^2}+\dfrac{3}{x^3-2x^2+x}$

$=\dfrac{x-1}{x^3}-\dfrac{x+1}{x^2(x-1)}+\dfrac{3}{x(x^2-2x+1)}$

$=\dfrac{x-1}{x^3}-\dfrac{x+1}{x^2(x-1)}+\dfrac{3}{x(x-1)^2}$

$=\dfrac{\left(x-1\right)^3}{x^3\left(x-1\right)^2}-\dfrac{\left(x+1\right)x\left(x-1\right)}{x^3\left(x-1\right)^2}+\dfrac{3x^2}{x^3\left(x-1\right)^2}$

 $=\dfrac{\left(x-1\right)^3-\left(x+1\right)x\left(x-1\right)+3x^2}{x^3\left(x-1\right)^2}$

 $=\dfrac{x^3-3x^2+3x-1-x^3+x+3x^2}{x^3\left(x-1\right)^2}$

 $=\dfrac{4x-1}{x^3\left(x-1\right)^2}$

d.Ta có:

$\dfrac{xy}{ab}+\dfrac{(x-a)(y-a)}{a(a-b)}-\dfrac{(x-b)(y-b)}{b(a-b)}$

$=\dfrac{xy(a-b)}{ab(a-b)}+\dfrac{b(x-a)(y-a)}{ab(a-b)}-\dfrac{a(x-b)(y-b)}{ab(a-b)}$

$=\dfrac{xy(a-b)+b(x-a)(y-a)-a(x-b)(y-b)}{ab(a-b)}$

$=\dfrac{xy(a-b)+b(xy-a(x+y)+a^2)-a(xy-b(x+y)+b^2)}{ab(a-b)}$

$=\dfrac{(axy-bxy)+(bxy-ab(x+y)+a^2b)-(axy-ab(x+y)+ab^2)}{ab(a-b)}$

$=\dfrac{axy-bxy+bxy-ab(x+y)+a^2b-axy+ab(x+y)-ab^2}{ab(a-b)}$

$=\dfrac{-ab^2+a^2b}{ab(a-b)}$

$=\dfrac{ab(a-b)}{ab(a-b)}$

$=1$

e.Ta có:

$\dfrac{x^3}{x-1}-\dfrac{x^2}{x+1}-\dfrac{1}{x-1}+\dfrac1{x+1}$

$=\dfrac{x^3-1}{x-1}-\dfrac{x^2-1}{x+1}$

$=\dfrac{(x-1)(x^2+x+1)}{x-1}-\dfrac{(x-1)(x+1)}{x+1}$

$=x^2+x+1-(x-1)$

$=x^2+x+1-x+1$

$=x^2+2$

f.Ta có:

$\dfrac{x^3+x^2-2x-20}{x^2-4}-\dfrac{5}{x+2}+\dfrac{3}{x-2}$

$=\dfrac{x^3+x^2-2x-20}{(x-2)(x+2)}-\dfrac{5(x-2)}{(x+2)(x-2)}+\dfrac{3(x+2)}{(x+2)(x-2)}$

$=\dfrac{x^3+x^2-2x-20-5(x-2)+3(x+2)}{(x+2)(x-2)}$

$=\dfrac{x^3+x^2-2x-20-5x+10+3x+6}{(x+2)(x-2)}$

$=\dfrac{x^3+x^2-4x-4}{(x+2)(x-2)}$

$=\dfrac{(x^3-4x)+(x^2-4)}{x^2-4}$

$=\dfrac{x(x^2-4)+(x^2-4)}{x^2-4}$

$=\dfrac{(x+1)(x^2-4)}{x^2-4}$

$=x+1$