Упр.21.36 ГДЗ Мерзляк Поляков 7 класс (Алгебра)

1) a^3+2a^2-3; 2) b^3+b^2+4; 3) x^3-7x-6;
4) a^3-2ab^2-b^3; 5) m^5+m^4+1; 6) x^8+x^4-2.

1. $$a^3+2a^2-3=a^3-a^2+3a^2-3=a^2(a-1)+3(a^2-1)$$
   $$=a^2(a-1)+3(a-1)(a+1)=(a-1)(a^2+3a+3).$$

2. $$b^3+b^2+4=b^3+2b^2-b^2+4=b^2(b+2)-(b^2-4)$$
   $$=b^2(b+2)-(b-2)(b+2)=(b+2)(b^2-b+2).$$

3. $$x^3-7x-6=x^3-x-6x-6=x(x^2-1)-6(x+1)$$
   $$=x(x-1)(x+1)-6(x+1)=(x+1)(x^2-x-6)$$
   $$=(x+1)(x-3)(x+2).$$

4. $$a^3-2ab^2-b^3=a^3-ab^2-ab^2-b^3=a(a^2-b^2)-b^2(a+b)$$
   $$=a(a-b)(a+b)-b^2(a+b)=(a+b)(a(a-b)-b^2)$$
   $$=(a+b)(a^2-ab-b^2).$$

5. $$m^5+m^4+1=m^5+m^4+m^3-m^3+1$$
   $$=m^3(m^2+m+1)-(m^3-1)$$
   $$=m^3(m^2+m+1)-(m-1)(m^2+m+1)$$
   $$=(m^2+m+1)(m^3-m+1).$$

6. $$x^8+x^4-2=x^8-1+x^4-1=(x^8-1)+(x^4-1)$$
   $$=(x^4-1)(x^4+1)+(x^4-1)=(x^4-1)(x^4+2)$$
   $$=(x^2-1)(x^2+1)(x^4+2)=(x-1)(x+1)(x^2+1)(x^4+2).$$

Ответ

1) $$\left(a-1\right)\left(a^2+3a+3\right);$$
2) $$\left(b+2\right)\left(b^2-b+2\right);$$
3) $$\left(x+1\right)\left(x-3\right)\left(x+2\right);$$
4) $$\left(a+b\right)\left(a^2-ab-b^2\right);$$
5) $$\left(m^2+m+1\right)\left(m^3-m+1\right);$$
6) $$\left(x-1\right)\left(x+1\right)\left(x^2+1\right)\left(x^4+2\right).$$