Рационални алгебарски изрази 4
Дистрибутивни закон. Разлика и збир кубова. Растављање алгебарских израза на чиниоце. Једноставни примери.
Задаци
Текст задатака објашњених у видео лекцији.
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) = $
$= \left( {x - 5y} \right)\left( {19{x^2} + 7{y^2} + 5xy} \right)$