\[\boxed{\text{345\ (345).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[y = x^{4} - ax^{3} - 10x^{2} + 80x - 96\]
\[y(4) = 0 \Longrightarrow 4^{4} - a \cdot 4^{3} -\]
\[- 10 \cdot 4^{2} + 80 \cdot 4 - 96 = 0.\]
\[256 - 64a - 160 + 320 -\]
\[- 96 = 0\]
\[- 64a = - 320\]
\[a = 5.\]
\[\Longrightarrow y = x^{4} - 5x^{3} - 10x^{2} +\]
\[+ 80x - 96.\]
\[x^{4} - 5x^{3} - 10x^{2} + 80x - 96 = 0;\ \ \]
\[x_{1} = 4 \Longrightarrow корень.\]
\[Схема\ Горнера:\]
\[1\] | \[- 5\] | \[- 10\] | \[80\] | \[- 96\] | |
---|---|---|---|---|---|
\[4\] | \[1\] | \[- 1\] | \[- 14\] | \[24\] | \[0\] |
\[\Longrightarrow x^{4} - 5x^{3} - 10x^{2} + 80x -\]
\[- 96 =\]
\[= (x - 4)\left( x^{3} - x^{2} - 14x + 24 \right);\]
\[x^{3} - x^{2} - 14x + 24 = 0.\]
\[Схема\ Горнера:\]
\[1\] | \[- 1\] | \[- 14\] | \[24\] | |
---|---|---|---|---|
\[3\] | \[1\] | \[2\] | \[- 8\] | \[0\] |
\[\Longrightarrow x^{4} - 5x^{3} - 10x^{2} + 80x -\]
\[- 96 =\]
\[= (x - 4)(x - 3)\left( x^{2} + 2x - 8 \right);\]
\[x^{2} + 2x - 8 = 0\]
\[D = 1 + 8 = 9\]
\[x = - 1 \pm 3,\ \ x = - 4;2.\]
\[Координаты\ точек\ \]
\[пересечения:(4;0),\ (3;0),\ \]
\[( - 4;0),\ (2;0).\]
\[\Longrightarrow a = 5.\]