[tex]\frac{2x-4}{x^{2}-2x+1} - \frac{x+2}{x}+\frac{x+1}{x-1}\\\\\\Dziedzina:\\x^{2}-2x+1 = (x-1)^{2}\\x-1 \neq 0 \ \ \wedge \ \ x \neq 0\\x \neq 1 \ \ \wedge \ \ x \neq 0\\D = R \setminus\{0,1\}\\\\\\\frac{2x-4}{x^{2}-2x+1} - \frac{x+2}{x} + \frac{x+1}{x-1} = \frac{x(2x-4)-(x+2)(x-1)^{2}+x(x-1)(x+1)}{x(x-1)^{2}}=\\\\\\=\frac{2x^{2}-4x-(x+2)(x^{2}-2x+1)+x(x^{2}-1)}{x(x-1)^{2}}=\frac{2x^{2}-4x-(x^{3}-2x^{2}+x+2x^{2}-4x+2)+x^{3}-x}{x(x-1)^{2}}=[/tex]
[tex]=\frac{2x^{2}-4x-(x^{3}-3x+2)+x^{3}-x}{x(x-1)^{2}} = \frac{2x^{2}-4x-x^{3}+3x-2+x^{3}-x}{x(x-1)^{2}} = \frac{2x^{2}-2x-2}{x(x-1)^{2}}[/tex]
Wykorzystano wzory skróconego mnożenia:
[tex](a-b)^{2} = a^{2}-2ab + b^{2}\\\\(a+b)(a-b) = a^{2}-b^{2}[/tex]
