Упр.21.13 ГДЗ Мерзляк Поляков 7 класс (Алгебра)

1) (a^2+b^2 )^2-4a^2 b^2; 5) 9a^2+c^2+6ac-9;
2) 81-(x^2+6x)^2; 6) a^2-b^2-10b-25;
3) a^2+2ab+b^2-c^2; 7) 49-y^2+x^2-14x;
4) c^2+4c+4-k^2; 8) mn^2-m^3-12m^2-36m.

1. $$\begin{aligned} (a^2+b^2)^2-4a^2b^2&=(a^2+b^2)^2-(2ab)^2\\ &=(a^2+b^2-2ab)(a^2+b^2+2ab)\\ &=(a-b)^2(a+b)^2. \end{aligned}$$
2. $$\begin{aligned} 81-(x^2+6x)^2&=9^2-(x^2+6x)^2\\ &=(9-(x^2+6x))(9+(x^2+6x))\\ &=(9-x^2-6x)(x^2+6x+9)\\ &=(9-x^2-6x)(x+3)^2. \end{aligned}$$
3. $$\begin{aligned} a^2+2ab+b^2-c^2&=(a+b)^2-c^2\\ &=(a+b-c)(a+b+c). \end{aligned}$$
4. $$\begin{aligned} c^2+4c+4-k^2&=(c+2)^2-k^2\\ &=(c+2-k)(c+2+k). \end{aligned}$$
5. $$\begin{aligned} 9a^2+c^2+6ac-9&=(3a+c)^2-3^2\\ &=(3a+c-3)(3a+c+3). \end{aligned}$$
6. $$\begin{aligned} a^2-b^2-10b-25&=a^2-(b^2+10b+25)\\ &=a^2-(b+5)^2\\ &=(a-b-5)(a+b+5). \end{aligned}$$
7. $$\begin{aligned} 49-y^2+x^2-14x&=(x^2-14x+49)-y^2\\ &=(x-7)^2-y^2\\ &=(x-7-y)(x-7+y). \end{aligned}$$
8. $$\begin{aligned} mn^2-m^3-12m^2-36m&=m(n^2-m^2-12m-36)\\ &=m\bigl(n^2-(m^2+12m+36)\bigr)\\ &=m\bigl(n^2-(m+6)^2\bigr)\\ &=m(n-m-6)(n+m+6). \end{aligned}$$

Ответ

1) $$(a-b)^2(a+b)^2$$
2) $$(9-x^2-6x)(x+3)^2$$
3) $$(a+b-c)(a+b+c)$$
4) $$(c+2-k)(c+2+k)$$
5) $$(3a+c-3)(3a+c+3)$$
6) $$(a-b-5)(a+b+5)$$
7) $$(x-7-y)(x-7+y)$$
8) $$m(n-m-6)(n+m+6)$$