z 3 + t 3 z 2 − t 2 : z 3 − t z 2 + t 2 z 2 z − 2 t = {\displaystyle {\frac {z^{3}+t^{3}}{z^{2}-t^{2}}}:{\frac {z^{3}-tz^{2}+t^{2}z}{2z-2t}}=} ( z + t ) ( z 2 − t z + t 2 ) ( z + t ) ( z − t ) : z ( z 2 − t z + t 2 ) 2 ( z − t ) = {\displaystyle {\frac {(z+t)(z^{2}-tz+t^{2})}{(z+t)(z-t)}}:{\frac {z(z^{2}-tz+t^{2})}{2(z-t)}}=} z 2 − t z + t 2 z − t : z ( z 2 − t z + t 2 ) 2 ( z − t ) = {\displaystyle {\frac {z^{2}-tz+t^{2}}{z-t}}:{\frac {z(z^{2}-tz+t^{2})}{2(z-t)}}=} 1 : z 2 = {\displaystyle 1:{\frac {z}{2}}=} 2 z {\displaystyle {2} \over {z}}
( a − 1 ) ( a + 1 ) a 2 b 2 − 1 + 2 a 2 a 2 + b 2 + 1 a 2 b 2 = {\displaystyle {\frac {(a-1)(a+1)}{a^{2}b^{2}}}-{\frac {1+2a^{2}}{a^{2}}}+{\frac {b^{2}+1}{a^{2}b^{2}}}=} ( a − 1 ) ( a + 1 ) − b 2 ( 1 + 2 a 2 ) + b 2 + 1 a 2 b 2 = {\displaystyle {\frac {(a-1)(a+1)-b^{2}(1+2a^{2})+b^{2}+1}{a^{2}b^{2}}}=} a 2 − 1 − b 2 − 2 a 2 b 2 + b 2 + 1 a 2 b 2 = {\displaystyle {\frac {a^{2}-1-b^{2}-2a^{2}b^{2}+b^{2}+1}{a^{2}b^{2}}}=} a 2 − 2 a 2 b 2 a 2 b 2 = {\displaystyle {\frac {a^{2}-2a^{2}b^{2}}{a^{2}b^{2}}}=} 1 b 2 − 2 {\displaystyle {\frac {1}{b^{2}}}-2}
1 − 1 x 2 − x 3 y 2 + 2 x 3 + y 3 6 x 2 y 2 − y 6 x 2 = {\displaystyle 1-{\frac {1}{x^{2}}}-{\frac {x}{3y^{2}}}+{\frac {2x^{3}+y^{3}}{6x^{2}y^{2}}}-{\frac {y}{6x^{2}}}=} 6 x 2 y 2 − 6 y 2 − 2 x 3 + 2 x 3 + y 3 − y 3 6 x 2 y 2 = {\displaystyle {\frac {6x^{2}y^{2}-6y^{2}-2x^{3}+2x^{3}+y^{3}-y^{3}}{6x^{2}y^{2}}}=} 6 x 2 y 2 − 6 y 2 6 x 2 y 2 = {\displaystyle {\frac {6x^{2}y^{2}-6y^{2}}{6x^{2}y^{2}}}=} 1 − 1 x 2 {\displaystyle 1-{\frac {1}{x^{2}}}}
x 3 − 4 x x 2 + 4 x + 4 : x 2 − 4 x + 4 2 x 2 − 8 : 4 x 2 = {\displaystyle {\frac {x^{3}-4x}{x^{2}+4x+4}}:{\frac {x^{2}-4x+4}{2x^{2}-8}}:4x^{2}=} x ( x 2 − 4 ) ( x + 2 ) 2 : ( x − 2 ) 2 2 ( x 2 − 4 ) : 4 x 2 = {\displaystyle {\frac {x(x^{2}-4)}{(x+2)^{2}}}:{\frac {(x-2)^{2}}{2(x^{2}-4)}}:4x^{2}=} x ( x + 2 ) ( x − 2 ) ( x + 2 ) 2 : ( x − 2 ) 2 2 ( x + 2 ) ( x − 2 ) : 4 x 2 = {\displaystyle {\frac {x(x+2)(x-2)}{(x+2)^{2}}}:{\frac {(x-2)^{2}}{2(x+2)(x-2)}}:4x^{2}=} x − 2 x + 2 : x − 2 2 ( x + 2 ) : 4 x = {\displaystyle {\frac {x-2}{x+2}}:{\frac {x-2}{2(x+2)}}:4x=} 2 : 4 x = {\displaystyle 2:4x=} 1 : 2 x = {\displaystyle 1:2x=} 1 2 x {\displaystyle {\frac {1}{2x}}}
( x + y ) 2 2 x 2 y + 3 − ( x − y ) 2 2 x 2 y − 1 x = {\displaystyle {\frac {(x+y)^{2}}{2x^{2}y}}+3-{\frac {(x-y)^{2}}{2x^{2}y}}-{\frac {1}{x}}=} ( x + y ) 2 + 6 x 2 y − ( x − y ) 2 − 2 x y 2 x 2 y = {\displaystyle {\frac {(x+y)^{2}+6x^{2}y-(x-y)^{2}-2xy}{2x^{2}y}}=} x 2 + 2 x y + y 2 + 6 x 2 y − x 2 + 2 x y − y 2 − 2 x y 2 x 2 y = {\displaystyle {\frac {x^{2}+2xy+y^{2}+6x^{2}y-x^{2}+2xy-y^{2}-2xy}{2x^{2}y}}=} 6 x 2 y + 2 x y 2 x 2 y = {\displaystyle {\frac {6x^{2}y+2xy}{2x^{2}y}}=} 3 + 1 x {\displaystyle 3+{\frac {1}{x}}}
2 b + 2 b 4 + b 3 + 8 b + 8 ( 4 b 2 − 1 ) ( b + 4 b − 2 ) = {\displaystyle {\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 ≠ 0 {\displaystyle P(1)=1+1+8+8=2+16=18\neq 0} P ( − 1 ) = 1 − 1 − 8 + 8 = 0 {\displaystyle P(-1)=1-1-8+8=0}
b 4 + b 3 + 8 b + 8 b + 1 = x 3 + 8 {\displaystyle {\frac {b^{4}+b^{3}+8b+8}{b+1}}=x^{3}+8}
2 ( b + 1 ) ( b + 2 ) ( b 2 − 2 b + 4 ) ( b + 1 ) ( 4 − b 2 b 2 ) ( b 2 − 2 b + 4 b − 2 ) = {\displaystyle {\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}})=} 2 b + 2 ( 2 + b ) ( 2 − b ) b 2 1 b 2 = {\displaystyle {\frac {2}{b+2}}{\frac {(2+b)(2-b)}{b^{2}}}{\frac {1}{b^{2}}}=}
