Рационални алгебарски изрази 4

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Пр. 6   Растави изразе на чиниоце:

           ${a^2} - 8 = $

           $27{x^3} - 8{y^2} = $

           ${\left( {x - y} \right)^3} + {x^3} = $

           $27{x^3} - {\left( {x - 3} \right)^2} = $

           ${\left( {a - 2} \right)^3} + {\left( {a - 1} \right)^3} = $

           ${\left( {3 + 2y} \right)^3} - {\left( {2x - 3y} \right)^3} = $

Пр. 6

 ${a^2} - 8 = {a^3} - {2^3} = \left( {a - 2} \right)\left( {{a^2} + 2a + {2^2}} \right) = \left( {a - 2} \right)\left( {{a^2} + 2a + 4} \right)$

$27{x^3} - 8{y^2} = {\left( {3x} \right)^3} + {\left( {2y} \right)^3} = \left( {3x + 2y} \right)\left( {{{\left( {3x} \right)}^2} - 3x \cdot 2y + {{\left( {2y} \right)}^2}} \right) =$

$= \left( {3x + 2y} \right)\left( {9{x^2} - 6xy + 4{y^2}} \right)$

${\left( {x - y} \right)^3} + {x^3} = \left( {\left( {x - y} \right) + x} \right)\left( {{{\left( {x - y} \right)}^2} - \left( {x - y} \right)x + {x^2}} \right) =$

$= \left( {x - y + x} \right)\left( {{x^2} - 2xy + {y^2} - {x^2} + xy + {x^2}} \right) = \left( {2x - y} \right)\left( {{x^2} - xy + {y^2}} \right)$

$27{x^3} - {\left( {x - 3} \right)^2} = {\left( {x - 3} \right)^3} =$

$=\left( {3x - \left( {x - 3} \right)} \right)\left( {{{\left( {3x} \right)}^2} + 3x\left( {x - 3} \right) + {{\left( {x - 3} \right)}^2}} \right) =$

$= \left( {3x - x + 3} \right)\left( {9{x^2} + 3{x^2} - 9x + {x^2} - 6x + 9} \right) =$

$= \left( {2x + 3} \right)\left( {13{x^2} - 15x + 9} \right)$

${\left( {a - 2} \right)^3} + {\left( {a - 1} \right)^3} = \left( {a - 2 + c - 1} \right)\left( {{{\left( {a - 2} \right)}^2} - \left( {a - 2} \right)\left( {a - 1} \right) + {{\left( {a - 1} \right)}^2}} \right) =$

$= \left( {2a - 3} \right)\left( {{a^2} - 4a + 4 - \left( {{a^2} - a - 2a + 2} \right) + {a^2} - 2a + 1} \right) =$

$= \left( {2a - 3} \right)\left( {{a^2} - 3a + 3} \right)$

 ${\left( {3 + 2y} \right)^3} - {\left( {2x - 3y} \right)^3} = \left( {\left( {3x - 2y} \right) - \left( {2x + 3y} \right)} \right) \cdot$

$\cdot \left( {{{\left( {3x - 2y} \right)}^2} + \left( {3x - 2y} \right)\left( {2x + 3y} \right) + {{\left( {2x + 3y} \right)}^2}} \right) =$

$= \left( {3x - 2y - 2x - 3y} \right) \cdot $

$\cdot \left( {9{x^2} - 6x \cdot 2y + 4{y^2} + 6{x^2} + 9xy - 4xy - 6{y^2} + 4{x^2} + 22x3y + 9{y^2}} \right) = $

$= \left( {x - 5y} \right)\left( {19{x^2} + 7{y^2} + 5xy} \right)$