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МатематикаРешебник по алгебре и начала математического анализа 10 класс Колягин Задание 399
Задание 399
\[\boxed{\mathbf{399}.}\]
\[1)\ x^{4} + 2x^{3} - x^{2} + 2x + 1 = 0\]
\[x^{4} + 3x^{3} - x^{3} + x^{2} + x^{2} -\]
\[- 3x^{2} - x + 3x + 1 = 0\]
\[x^{4} + x^{2}(3x + 1) - x^{2}(x - 1) -\]
\[- x(3x + 1) + (3x + 1) = 0\]
\[x^{2}\left( x^{2} - x + 1 \right) +\]
\[+ (3x + 1)\left( x^{2} - x + 1 \right) = 0\]
\[\left( x^{2} - x + 1 \right)\left( x^{2} + 3x + 1 \right) = 0\]
\[x^{2} - x + 1 = 0\]
\[D = 1 - 4 < 0\]
\[корней\ нет.\]
\[x^{2} + 3x + 1 = 0\]
\[D = 9 - 4 = 5\]
\[x = \frac{- 3 \pm \sqrt{5}}{2}.\]
\[Ответ:x = \frac{- 3 \pm \sqrt{5}}{2}.\]
\[2)\ 2x^{4} + x^{3} - 10x^{2} - x + 2 = 0\]
\[Делители:\ \pm 1;\ \pm 2;\ \pm \frac{1}{2}.\]
| \[2\] | \[1\] | \[- 10\] | \[- 1\] | \[2\] | |
|---|---|---|---|---|---|
| \[2\] | \[2\] | \[5\] | \[0\] | \[- 1\] | \[0\] |
| \[- 0,5\] | \[2\] | \[4\] | \[- 2\] | \[0\] |
\[P(x) = (x - 2)(x + 0,5)\left( 2x^{2} + 4x - 2 \right) = 0\]
\[2x^{2} + 4x - 2 = 0\ \ \ |\ :2\]
\[x^{2} + 2x - 1 = 0\]
\[D_{1} = 1 + 1 = 2\]
\[x = - 1 \pm \sqrt{2}.\]
\[Ответ:x = - 0,5;2;\ - 1 \pm \sqrt{2}.\]
\[3)\ (x - 1)x(x + 1)(x + 2) = 24\]
\[\left( x^{2} - 1 \right)\left( x^{2} + 2x \right) - 24 = 0\]
\[x^{4} + 2x^{3} - x^{2} - 2x - 24 = 0\]
\[Делители:\ \pm 1;\ \pm 2;\ \pm 3;\ \pm 4;\ \]
\[\pm 6;\ \pm 8;\ \pm 12;\ \pm 24.\]
| \[1\] | \[2\] | \[- 1\] | \[- 2\] | \[- 24\] | |
|---|---|---|---|---|---|
| \[- 3\] | \[1\] | \[- 1\] | \[2\] | \[- 8\] | \[0\] |
| \[2\] | \[1\] | \[1\] | \[4\] | \[0\] |
\[P(x) = (x + 3)(x - 2)\left( x^{2} + x + 4 \right) = 0\]
\[x^{2} + x + 4 = 0\]
\[D = 1 - 16 < 0\]
\[нет\ корней.\]
\[Ответ:x = - 3;\ \ 2.\]
\[x^{2} + 5x = - 7\]
\[x^{2} + 5x + 7 = 0\]
\[D = 25 - 28 < 0\]
\[нет\ корней.\]
\[Ответ:x = \frac{- 5 \pm \sqrt{13}}{2}.\]
