Question

Упростите номера 40,41 и 42
Даю 30 баллов

Answer 1

$\displaystyle\bf\\40\\\\\frac{y^{2} -x^{2} }{2xy} :\frac{x+y}{2x} =\frac{(y-x)\cdot(y+x)}{2xy}\cdot\frac{2x}{x+y} =\frac{y-x}{y} =\\\\\\=\frac{y}{y} -\frac{x}{y} =\boxed{1-\frac{x}{y} }\\\\41)\\\\\frac{x^{3} +x^{2} +x+1}{x^{2} +1} +x=\frac{(x^{3}+x)+(x^{2}+1) }{x^{2} +1} +x=\\\\\\=\frac{x\cdot(x^{2}+1)+(x^{2}+1) }{x^{2} +1} +x=\frac{(x^{2} +1)\cdot(x+1)}{x^{2} +1}+x=\\\\\\=x+1+x=\boxed{2x+1}\\\\42)\\\\\frac{x^{3} +2x^{2} +x}{(x+1)^{2} } -x=\frac{x\cdot(x^{2} +2x+1)}{(x+1)^{2} } -x=$

$\displaystyle\bf\\=\frac{x\cdot(x+1)^{2} }{(x+1)^{2} } -x=x-x=\boxed{0}$