Страница 335 номер 1113, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение
Записать разложение бинома: 1) \((3a + 1)^5\); 2) \((x + 3)^6\); 3) \(\left(x - \frac{1}{3}\right)^7\); 4) \(\left(\frac{a}{3} - 1\right)^5\); 5) \((10x - 0{,}1)^6\); 6) \((0{,}1b - 10)^7\); 7) \(\left(\frac{2}{a} + \frac{a}{2}\right)^7\); 8) \(\left(\frac{2}{c} + \frac{c}{2}\right)^8\).
1) \((3a + 1)^{5} = (3a)^{5} + 5(3a)^{4} + 10(3a)^{3} + 10(3a)^{2} + 5(3a) + 1 =\)
\(= 243a^{5} + 405a^{4} + 270a^{3} + 90a^{2} + 15a + 1;\)
2) \((x + 3)^{6} = x^{6} + 6x^{5} \cdot 3 + 15x^{4} \cdot 9 + 20x^{3} \cdot 27 + 15x^{2} \cdot 81 +\)
\(+ 6x \cdot 243 + 729 =\)
\(= x^{6} + 18x^{5} + 135x^{4} + 540x^{3} + 1215x^{2} + 1458x + 729;\)
3) \(\left( x - \frac{1}{3} \right)^{7} = x^{7} - \frac{7x^{6}}{3} + \frac{21x^{5}}{9} - \frac{35x^{4}}{27} + \frac{35x^{3}}{81} - \frac{21x^{2}}{243} + \frac{7x}{729} - \frac{1}{2187}\);
4) \(\left( \frac{a}{3} - 1 \right)^{5} = \frac{a^{5}}{243} - \frac{5a^{4}}{81} + \frac{10a^{3}}{27} - \frac{10a^{2}}{9} + \frac{5a}{3} - 1\);
5) \((10x - 0{,}1)^{6} = (10x)^{6} - \frac{6(10x)^{5}}{10} + \frac{15(10x)^{4}}{10^{2}} - \frac{20(10x)^{3}}{10^{3}} + \frac{15(10x)^{2}}{10^{4}} -\)
\(- \dfrac{6(10x)}{10^{5}} + \dfrac{1}{10^{6}} =\)
\(= 1\ 000\ 000x^{6} - 60000x^{5} + 1500x^{4} - 20x^{3} + \dfrac{15x^{2}}{100} - \dfrac{6x}{10^{4}} + \dfrac{1}{10^{6}};\)
6) \((0{,}1b - 10)^{7} = \left( \frac{b}{10} \right)^{7} - \frac{7 \cdot 10b^{6}}{10^{6}} + \frac{21 \cdot 10^{2}b^{5}}{10^{5}} - \frac{35 \cdot 10^{3}b^{4}}{10^{4}} + \frac{35 \cdot 10^{4}b^{3}}{10^{3}} - \frac{21 \cdot 10^{5}b^{2}}{10^{2}} +\)
\(+ \dfrac{7 \cdot 10^{6}b}{10} - 10^{7} =\)
\(= \dfrac{b^{7}}{10^{7}} - \dfrac{7b^{6}}{10^{5}} + \dfrac{21b^{5}}{10^{3}} - \dfrac{35b^{4}}{10} + 350b^{3} - 21000b^{2} + 7b \cdot 10^{5} - 10^{7};\)
7) \(\left( \frac{2}{a} + \frac{a}{2} \right)^{7} = \left( \frac{2}{a} \right)^{7} + 7\left( \frac{2}{a} \right)^{6}\left( \frac{a}{2} \right) + 21\left( \frac{2}{a} \right)^{5}\left( \frac{a}{2} \right)^{2} + 35\left( \frac{2}{a} \right)^{4}\left( \frac{a}{2} \right)^{3} +\)
\(+ 35\left( \dfrac{2}{a} \right)^{3}\left( \dfrac{a}{2} \right)^{4} + 21\left( \dfrac{2}{a} \right)^{2}\left( \dfrac{a}{2} \right)^{5} + 7\left( \dfrac{2}{a} \right)\left( \dfrac{a}{2} \right)^{6} + \left( \dfrac{a}{2} \right)^{7} =\)
\(= \left( \dfrac{2}{a} \right)^{7} + 7\left( \dfrac{2}{a} \right)^{5} + 21\left( \dfrac{2}{a} \right)^{3} + 35\left( \dfrac{2}{a} \right) +\)
\(+ 35\left( \dfrac{a}{2} \right) + 21\left( \dfrac{a}{2} \right)^{3} + 7\left( \dfrac{a}{2} \right)^{5} + \left( \dfrac{a}{2} \right)^{7} =\)
\(= \dfrac{128}{a^{7}} + \dfrac{224}{a^{5}} + \dfrac{168}{a^{3}} + \dfrac{70}{a} + \dfrac{35a}{2} + \dfrac{21a^{3}}{8} + \dfrac{7a^{5}}{32} + \dfrac{a^{7}}{128};\)
8) \(\left( \frac{2}{c} + \frac{c}{2} \right)^{8} = \left( \frac{2}{c} \right)^{8} + 8\left( \frac{2}{c} \right)^{6} + 28\left( \frac{2}{c} \right)^{4} + 56\left( \frac{2}{c} \right)^{2} + 70 + 56\left( \frac{c}{2} \right)^{2} +\)
\(+ 28\left( \dfrac{c}{2} \right)^{4} + 8\left( \dfrac{c}{2} \right)^{6} + \left( \dfrac{c}{2} \right)^{8} =\)
\(= \dfrac{256}{c^{8}} + \dfrac{512}{c^{6}} + \dfrac{448}{c^{4}} + \dfrac{224}{c^{2}} + 70 + 14c^{2} + \dfrac{7c^{4}}{4} + \dfrac{c^{6}}{8} + \dfrac{c^{8}}{256}.\)
