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МатематикаРешебник по алгебре и начала математического анализа 10 класс Колягин Задание 346
Задание 346
\[\boxed{\mathbf{346}.}\]
\[1)\ P(x,\ y) = 15x^{4} - 8x^{3}y +\]
\[+ 31x^{2}y^{2} - 16xy^{3} + 2y^{4}\]
\[y = tx:\]
\[P(x,\ y) = 15x^{4} - 8tx^{4} +\]
\[+ 31t^{2}x^{4} - 16t^{3}x^{4} + 2t^{4}x^{4} =\]
\[= x^{4}\left( 2t^{4} - 16t^{3} + 31t^{2} - 8t + 15 \right).\]
| \[2\] | \[- 16\] | \[31\] | \[- 8\] | \[15\] | |
|---|---|---|---|---|---|
| \[3\] | \[2\] | \[- 10\] | \[1\] | \[- 5\] | \[0\] |
| \[5\] | \[2\] | \[0\] | \[1\] | \[0\] |
\[Q(x,\ y) =\]
\[= x^{4}(t - 3)(t - 5)\left( 2t^{2} + 1 \right).\]
\[Делаем\ обратную\ замену:\]
\[\text{\ \ \ }t = \frac{y}{x}.\]
\[P(x,\ y) =\]
\[= x^{4}\left( \frac{y}{x} - 3 \right)\left( \frac{y}{x} - 5 \right)\]
\[\left( 2 \cdot \frac{y^{2}}{x^{2}} + 1 \right) =\]
\[= (y - 3x)(y - 5x)\left( 2y^{2} + x^{2} \right).\]
\[2)\ P(x,\ y) = 12x^{5} - 32x^{4}y +\]
\[+ 9x^{3}y^{2} + 16x^{2}y^{3} -\]
\[- 3xy^{4} - 2y^{5}.\]
\[y = tx:\]
\[P(x,\ y) = 12x^{5} - 32x^{4}tx +\]
\[+ 9x^{3}\left( \text{tx} \right)^{2} + 16x^{2}\left( \text{tx} \right)^{3} -\]
\[- 3x{(tx)}^{4} - 2{(tx)}^{5}.\]
\[= 12x^{5} - 32xt^{5} + 9t^{2}x^{5} +\]
\[+ 16t^{3}x^{5} - 3t^{4}x^{5} - 2t^{5}x^{5} =\]
\[= x^{5}\left( - 2t^{5} - 3t^{4} + 16t^{3} + 9t^{2} - 32t + 12 \right).\]
| \[- 2\] | \[- 3\] | \[16\] | \[9\] | \[- 32\] | \[12\] | |
|---|---|---|---|---|---|---|
| \[1\] | \[- 2\] | \[- 5\] | \[11\] | \[20\] | \[- 12\] | \[0\] |
| \[- 3\] | \[- 2\] | \[1\] | \[8\] | \[- 4\] | \[0\] | |
| \[2\] | \[- 2\] | \[- 3\] | \[2\] | \[0\] | ||
| \[- 2\] | \[- 2\] | \[1\] | \[0\] |
\[Q(x,\ y) = (t - 1)(t + 3)\]
\[(t - 2)(t + 2)( - 2t + 1).\]
\[Делаем\ обратную\ замену\ \]
\[t = \frac{y}{x}:\]
\[P(x,\ y) = x^{5}\left( \frac{y}{x} - 1 \right)\left( \frac{y}{x} + 3 \right)\]
\[\left( \frac{y}{x} - 2 \right)\left( \frac{y}{x} + 2 \right)\left( - \frac{2y}{x} + 1 \right) =\]
\[= (y - x)(y + 3x)(y - 2x)\]
\[(y + 2x)( - 2y + x).\]
