Упр.1092 ГДЗ Алимов 10-11 класс (Алгебра)

1. Записать разложение бинома: 1) $$(1+x)^8$$; 2) $$(x+1)^7$$; 3) $$(a-1)^9$$; 4) $$(y-1)^{10}$$; 5) $$(2x+1)^5$$; 6) $$(x+2)^6$$; 7) $$(3x+2)^4$$; 8) $$(2a+3)^5$$; 9) $$(2a-\frac{1}{2})^5$$; 10) $$(3x-\frac{1}{3})^4$$.

Используем формулу бинома Ньютона:

$$$(a+b)^n=C_n^0a^n+C_n^1a^{n-1}b+C_n^2a^{n-2}b^2+\dots+C_n^nb^n$$$

1. $$$(1+x)^8=1+8x+28x^2+56x^3+70x^4+56x^5+28x^6+8x^7+x^8$$$

2. $$$(x+1)^7=x^7+7x^6+21x^5+35x^4+35x^3+21x^2+7x+1$$$

3. $$$(a-1)^9=a^9-9a^8+36a^7-84a^6+126a^5-126a^4+84a^3-36a^2+9a-1$$$

4. $$$(y-1)^{10}=y^{10}-10y^9+45y^8-120y^7+210y^6-252y^5+210y^4-120y^3+45y^2-10y+1$$$

5. $$$(2x+1)^5=32x^5+80x^4+80x^3+40x^2+10x+1$$$

6. $$$(x+2)^6=x^6+12x^5+60x^4+160x^3+240x^2+192x+64$$$

7. $$$(3x+2)^4=81x^4+216x^3+216x^2+96x+16$$$

8. $$$(2a+3)^5=32a^5+240a^4+720a^3+1080a^2+810a+243$$$

9. $$$(2a-\tfrac12)^5=32a^5-40a^4+20a^3-5a^2+\tfrac58a-\tfrac1{32}$$$

10. $$$(3x-\tfrac13)^4=81x^4-36x^3+6x^2-\tfrac49x+\tfrac1{81}$$$

Ответ:

1. $$$(1+x)^8=1+8x+28x^2+56x^3+70x^4+56x^5+28x^6+8x^7+x^8$$$
2. $$$(x+1)^7=x^7+7x^6+21x^5+35x^4+35x^3+21x^2+7x+1$$$
3. $$$(a-1)^9=a^9-9a^8+36a^7-84a^6+126a^5-126a^4+84a^3-36a^2+9a-1$$$
4. $$$(y-1)^{10}=y^{10}-10y^9+45y^8-120y^7+210y^6-252y^5+210y^4-120y^3+45y^2-10y+1$$$
5. $$$(2x+1)^5=32x^5+80x^4+80x^3+40x^2+10x+1$$$
6. $$$(x+2)^6=x^6+12x^5+60x^4+160x^3+240x^2+192x+64$$$
7. $$$(3x+2)^4=81x^4+216x^3+216x^2+96x+16$$$
8. $$$(2a+3)^5=32a^5+240a^4+720a^3+1080a^2+810a+243$$$
9. $$$(2a-\tfrac12)^5=32a^5-40a^4+20a^3-5a^2+\tfrac58a-\tfrac1{32}$$$
10. $$$(3x-\tfrac13)^4=81x^4-36x^3+6x^2-\tfrac49x+\tfrac1{81}$$$