Упр.337 ГДЗ Никольский Потапов 7 класс (Алгебра)

1. а)

   $$a(b-c)+b(c-a)+c(a-b)=ab-ac+bc-ab+ac-bc=0.$$

2. б)

   $$ab(c-d)-cd(a-b)-ac(b-d)-bd(c-a)=abc-abd-acd+bcd-abc+acd-bcd+abd=0.$$

3. в)

   $$\begin{aligned} &(m-n)(2m+3n)(m-7)+7(2m^2+2mn-3n^2) \\ &= (2m^2+mn-3n^2)(m-7)+14m^2+14mn-21n^2 \\ &= 2m^3-14m^2+m^2n-7mn-3mn^2+21n^2+14m^2+14mn-21n^2 \\ &= 2m^3+m^2n-3mn^2+7mn \\ &= m(2m^2+mn-3n^2+7n). \end{aligned}$$

4. г)

   $$\begin{aligned} &(a^3b-b^2)(a^2-2b)(a-3b)+3a^2b^2(a^3-2ab-b)+2b^2(a^4-ab+3b^2) \\ &= (a^5b-2a^3b^2-a^2b^2+2b^3)(a-3b)+3a^5b^2-6a^3b^3-3a^2b^3 \\ &\quad +2a^4b^2-2ab^3+6b^4 \\ &= a^6b-3a^5b^2-2a^4b^2+6a^3b^3-a^3b^2+3a^2b^3+2ab^3-6b^4 \\ &\quad +3a^5b^2-6a^3b^3-3a^2b^3+2a^4b^2-2ab^3+6b^4 \\ &= a^6b-a^3b^2 \\ &= a^3b(a^3-b). \end{aligned}$$

5. д)

   $$\begin{aligned} &(a^2-4a+4)(a^2+4a+4)-a^2(a^2-8) \\ &= (a^4+4a^3+4a^2-4a^3-16a^2-16a+4a^2+16a+16)-a^4+8a^2 \\ &= 16. \end{aligned}$$

6. е)

   $$\begin{aligned} &(4a^2+4a+1)(4a^2-4a+1)-8a^2(2a^2-1) \\ &= (16a^4-16a^3+4a^2+16a^3-16a^2+4a+4a^2-4a+1)-16a^4+8a^2 \\ &= 1. \end{aligned}$$

7. ж)

   $$\begin{aligned} &(a-1)(a+1)(a^2+1)(a^4+1)-a^8 \\ &= (a^2-1)(a^2+1)(a^4+1)-a^8 \\ &= (a^4-1)(a^4+1)-a^8 \\ &= a^8-1-a^8 \\ &= -1. \end{aligned}$$

8. з)

   $$\begin{aligned} &(a-2)(a+2)(a^2+4)(a^4+16)-a^8 \\ &= (a^2-4)(a^2+4)(a^4+16)-a^8 \\ &= (a^4-16)(a^4+16)-a^8 \\ &= a^8-256-a^8 \\ &= -256. \end{aligned}$$

Ответ

а) $$0$$; б) $$0$$; в) $$m(2m^2+mn-3n^2+7n)$$; г) $$a^3b(a^3-b)$$; д) $$16$$; е) $$1$$; ж) $$-1$$; з) $$-256$$.