Упростить выражение:[latex] \frac{ x^{3}- x^{2} -x+1 }{ x^{4}- x^{3}-3 x^{2} +5x-2 } [/latex]Ответ: [latex] \frac{x+1}{ x^{2} +x-2} [/latex]

[latex]= \frac{ x^{2} (x-1)-(x-1) }{ x^{4} - x^{3} -3 x^{2} +3x+2x-2} = \\ = \frac{(x-1)( x^{2} -1)}{ x^{3}(x-1)-3x(x-1)+2(x-1) } =\\ = \frac{(x-1)( x^{2} -1)}{(x-1)( x^{3}-3x+2 )}= \\ = \frac{ x^{2} -1}{ x^{3}-3x+2} = \\ = \frac{(x-1)(x+1)}{ x^{3}-x-2x+2 }= \\ = \frac{(x-1)(x+1)}{(x-1)( x^{2} +x-2)} =\\= \frac{x+1}{ x^{2} +x-2} [/latex]

(x-1)(x2-1)/(x-1)(x3-3x+2)=x2-1/x3-3x+2=(x+1)(x-1)/x3-x-2x+2=(x-1)(x+1)/(x-1)(x2-x-2)=x+1/x2+x-2 Ответ:x+1/x2+x-2