Решебник по алгебре 7 класс Мерзляк Задание 739

\[\boxed{\text{739\ (739).\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[1)\ x^{4} - 5x^{2} + 4 =\]

\[= \left( x^{4} - 5x^{2} + 6,25 \right) - 2,25 =\]

\[= \left( x^{2} - 2,5 \right)^{2} - 2,25 =\]

\[= \left( x^{2} - 2,5 - 1,5 \right)\left( x^{2} - 2,5 + 1,5 \right) =\]

\[= \left( x^{2} - 4 \right)\left( x^{2} - 1 \right) =\]

\[= (x - 2)(x + 2)(x - 1)(x + 1)\]

\[2)\ x^{4} + x² + 1 =\]

\[= \left( x^{4} + 2x^{2} + 1 \right) - x^{2} =\]

\[= \left( x^{2} + 1 \right)^{2} - x^{2} =\]

\[= \left( x^{2} + 1 + x \right)\left( x^{2} + 1 - x \right)\]

\[3)\ 4x^{4} - 12x^{2} + 1 =\]

\[= \left( 4x^{4} + 4x^{2} + 1 \right) - 16x^{2} =\]

\[= \left( 2x^{2} + 1 \right)^{2} - (4x)^{2} =\]

\[= (2x^{2} + 1 - 4x)(2x^{2} + 1 + 4x)\]

\[4)\ x^{5} + x + 1 =\]

\[= \left( x^{5} - x^{2} \right) + \left( x^{2} + x + 1 \right) =\]

\[= x^{2}\left( x^{3} - 1 \right) + \left( x^{2} + x + 1 \right) =\]

\[= \left( x^{2} + x + 1 \right)\left( x^{2}(x - 1) + 1 \right) =\]

\[= (x^{2} + x + 1)(x^{3} - x^{2} + 1)\]

\[5)\ x^{4} + 4 =\]

\[= \left( x^{4} + 4x^{2} + 4 \right) - 4x^{2} =\]

\[= \left( x^{2} + 2 \right)^{2} - 4x^{2} =\]

\[= \left( x^{2} + 2 + 2x \right)\left( x^{2} + 2 - 2x \right)\]

\[6)\ x^{8} + x^{4} - 2 =\]

\[= \left( x^{8} - 2x^{4} + 1 \right) + \left( 3x^{4} - 3 \right) =\]

\[= \left( x^{4} - 1 \right)^{2} + 3\left( x^{4} - 1 \right) =\]

\[= \left( x^{4} - 1 \right)\left( x^{4} - 1 + 3 \right) =\]

\[= \left( x^{4} - 1 \right)\left( x^{4} + 2 \right) =\]

\[= \left( x^{2} - 1 \right)\left( x^{2} + 1 \right)\left( x^{4} + 2 \right) =\]

\[= (x - 1)(x + 1)(x^{2} + 1)(x^{4} + 2)\]