Упр.1113 ГДЗ Алимов 10-11 класс (Алгебра)

Записать разложение бинома:

1. $$\left(3a+1\right)^5$$;
2. $$\left(x+6\right)^6$$;
3. $$\left(x+\frac{1}{3}\right)^7$$;
4. $$\left(\frac{a}{3}-1\right)^5$$;
5. $$\left(10x-0{,}1\right)^6$$;
6. $$\left(0{,}1b-10\right)^7$$;
7. $$\left(\frac{2}{a}+\frac{a}{2}\right)^7$$;
8. $$\left(\frac{2}{c}+\frac{c}{2}\right)$$.

Используем биномиальную формулу Ньютона:

$$(a+b)^n=\sum_{k=0}^{n} C_n^k a^{n-k}b^k$$

1. $$(3a+1)^5=243a^5+405a^4+270a^3+90a^2+15a+1$$

2. $$(x+6)^6=x^6+36x^5+540x^4+4320x^3+19440x^2+46656x+46656$$

3. $$\left(x+\frac13\right)^7=x^7+\frac73x^6+\frac73x^5+\frac{35}{27}x^4+\frac{35}{81}x^3+\frac{7}{243}x^2+\frac{7}{2187}x+\frac{1}{2187}$$

4. $$\left(\frac a3-1\right)^5=\frac{1}{243}a^5-\frac{5}{81}a^4+\frac{10}{27}a^3-\frac{10}{9}a^2+\frac53a-1$$

5. $$(10x-0{,}1)^6=1000000x^6-600000x^5+150000x^4-20000x^3+1500x^2-60x+0{,}000001$$

6. $$(0{,}1b-10)^7=\frac{b^7}{10^7}-\frac{7b^6}{10^5}+\frac{21b^5}{10^3}-\frac{35b^4}{10}+350b^3-21000b^2+700000b-10000000$$

7. $$\left(\frac{2}{a}+\frac{a}{2}\right)^7=\frac{128}{a^7}+\frac{224}{a^5}+\frac{168}{a^3}+\frac{70}{a}+\frac{35}{2}a+\frac{21}{8}a^3+\frac{7}{32}a^5+\frac{1}{128}a^7$$

8. $$\left(\frac{2}{c}+\frac{c}{2}\right)^8=\frac{256}{c^8}+\frac{512}{c^6}+\frac{448}{c^4}+\frac{224}{c^2}+70+14c^2+\frac{7}{4}c^4+\frac{1}{8}c^6+\frac{1}{256}c^8$$