Question

Se considera expresia E(x) = ( $\frac{x}{x+1}$ - $\frac{2}{1-x} }$ ) : $\frac{x^{2} + x + 2 }{x^{4} - x^{2} } } }$
unde x∈ R \{ -1 , 0 , 1 }
Demonstrati ca E(x) = $x^{2}$

Answer 1

$( \frac{x}{x+1} - \frac{2}{1-x} ): \frac{ x^{2} +x+2}{x^{4}- x^{2} } =( \frac{x}{x+1} + \frac{2}{x-1} )* \frac{x^{4}- x^{2} }{ x^{2} +x+2}= \frac{x(x-1)+2(x+1)}{(x-1)(x+1)}* \\ *\frac{ x^{2} ( x^{2} -1)}{ x^{2} +x+2}= \frac{ x^{2} -x+2x+2}{ x^{2} -1} * \frac{ x^{2} ( x^{2} -1)}{ x^{2} +x+2}= \frac{ x^{2} +x+2}{1} * \frac{ x^{2} }{ x^{2} +x+2}= x^{2}$
Ceea ce trebuia de demonstrat