Користувач:Vogand/Дробові вирази

${\frac {z^{3}+t^{3}}{z^{2}-t^{2}}}:{\frac {z^{3}-tz^{2}+t^{2}z}{2z-2t}}=$
${\frac {(z+t)(z^{2}-tz+t^{2})}{(z+t)(z-t)}}:{\frac {z(z^{2}-tz+t^{2})}{2(z-t)}}=$
${\frac {z^{2}-tz+t^{2}}{z-t}}:{\frac {z(z^{2}-tz+t^{2})}{2(z-t)}}=$
$1:{\frac {z}{2}}=$
${2} \over {z}$

${\frac {(a-1)(a+1)}{a^{2}b^{2}}}-{\frac {1+2a^{2}}{a^{2}}}+{\frac {b^{2}+1}{a^{2}b^{2}}}=$
${\frac {(a-1)(a+1)-b^{2}(1+2a^{2})+b^{2}+1}{a^{2}b^{2}}}=$
${\frac {a^{2}-1-b^{2}-2a^{2}b^{2}+b^{2}+1}{a^{2}b^{2}}}=$
${\frac {a^{2}-2a^{2}b^{2}}{a^{2}b^{2}}}=$
${\frac {1}{b^{2}}}-2$

$1-{\frac {1}{x^{2}}}-{\frac {x}{3y^{2}}}+{\frac {2x^{3}+y^{3}}{6x^{2}y^{2}}}-{\frac {y}{6x^{2}}}=$
${\frac {6x^{2}y^{2}-6y^{2}-2x^{3}+2x^{3}+y^{3}-y^{3}}{6x^{2}y^{2}}}=$
${\frac {6x^{2}y^{2}-6y^{2}}{6x^{2}y^{2}}}=$
$1-{\frac {1}{x^{2}}}$

${\frac {x^{3}-4x}{x^{2}+4x+4}}:{\frac {x^{2}-4x+4}{2x^{2}-8}}:4x^{2}=$
${\frac {x(x^{2}-4)}{(x+2)^{2}}}:{\frac {(x-2)^{2}}{2(x^{2}-4)}}:4x^{2}=$
${\frac {x(x+2)(x-2)}{(x+2)^{2}}}:{\frac {(x-2)^{2}}{2(x+2)(x-2)}}:4x^{2}=$
${\frac {x-2}{x+2}}:{\frac {x-2}{2(x+2)}}:4x=$
$2:4x=$
$1:2x=$
${\frac {1}{2x}}$

${\frac {(x+y)^{2}}{2x^{2}y}}+3-{\frac {(x-y)^{2}}{2x^{2}y}}-{\frac {1}{x}}=$
${\frac {(x+y)^{2}+6x^{2}y-(x-y)^{2}-2xy}{2x^{2}y}}=$
${\frac {x^{2}+2xy+y^{2}+6x^{2}y-x^{2}+2xy-y^{2}-2xy}{2x^{2}y}}=$
${\frac {6x^{2}y+2xy}{2x^{2}y}}=$
$3+{\frac {1}{x}}$

${\frac {2b+2}{b^{4}+b^{3}+8b+8}}({\frac {4}{b^{2}}}-1)(b+{\frac {4}{b-2}})=$

Правило Руффіні:
$P(1)=1+1+8+8=2+16=18\neq 0$
$P(-1)=1-1-8+8=0$

${\frac {b^{4}+b^{3}+8b+8}{b+1}}=x^{3}+8$

${\frac {2(b+1)}{(b+2)(b^{2}-2b+4)(b+1)}}({\frac {4-b^{2}}{b^{2}}})({\frac {b^{2}-2b+4}{b-2}})=$
${\frac {2}{b+2}}{\frac {(2+b)(2-b)}{b^{2}}}{\frac {1}{b^{2}}}=$