Wykonaj działania i podaj konieczne zalożenia: a) [latex] frac{x+2}{x-1} - frac{4}{x^2-2x+1} - frac{x+1}{x} [/latex] b) [latex] frac{x^2-1}{x^2-5x} - frac{x^2-3}{x^2-10x+25} - frac{1}{5-x} [/latex] c) [latex](3 - frac{x}{x+2}) : (x+3)[/latex]

[latex]a) \\ frac{x+2}{x-1} - frac{4}{x^2-2x+1} - frac{x+1}{x}=frac{x+2}{x-1} - frac{4}{(x -1)^2} - frac{x+1}{x} \\ x-1 eq 0 wedge x eq 0\\ x eq 1 wedge x eq 0\\D=Rleft { 0,1 ight }[/latex] [latex]frac{x+2}{x-1} - frac{4}{(x -1)^2} - frac{x+1}{x}= frac{(x+2)(x-1)x-4x- (x-1)^2(x+1)}{x(x-1)^2}=\\= frac{(x^2-x+2 x-2)x-4x -(x +1)(x^2-2x+1) }{x(x-1)^2}=\\=frac{ x^3-x^2+2 x^2-2 x-4x - (x^3-2x^2+x+x^2-2x+1) }{x(x-1)^2}=\\= frac{ x^3-x^2+2 x^2-2 x-4x - x^3+2x^2-x-x^2+2x-1 }{x(x-1)^2}= frac{ 2x^2-5x -1 }{x(x-1)^2}[/latex] [latex]b)\\ frac{x^2-1}{x^2-5x} - frac{x^2-3}{x^2-10x+25} - frac{1}{5-x}= frac{x^2-1}{x(x -5 )} - frac{x^2-3}{(x -5)^2} + frac{1}{ x-5} \\x eq 0 wedge x-5 eq 0\\ x eq 0 wedge x eq 5\\D=Rsetminus left { 0,5 ight }[/latex] [latex]frac{x^2-1}{x(x -5 )} - frac{x^2-3}{(x -5)^2} + frac{1}{ x-5}= frac{(x-5)(x^2-1)-x(x^2-3) +x(x-5)}{x(x -5 )^2}=\\= frac{ x^3-x-5 x^2+5- x^3+3x + x^2-5x}{x(x -5 )^2}=frac{ -4x^2 -3x+5}{x(x -5 )^2}=- frac{ 4x^2+3x-5}{x(x -5 )^2}[/latex] [latex]c)\\ (3 - frac{x}{x+2}) : (x+3)\\x+2 eq 0 wedge x+3 eq 0\\x eq -2 wedge x eq -3\\D=Rsetminus left { -2,-3 ight }[/latex] [latex](3 - frac{x}{x+2}) : (x+3)= frac{3(x+2)-x}{x+2} *frac{1}{x+3} = frac{3 x+6-x}{x+2} *frac{1}{x+3} = \\= frac{2x+6 }{x+2} *frac{1}{x+3} = frac{2(x+3) }{x+2} *frac{1}{x+3} = frac{2 }{x+2}[/latex]