Упр.511 ГДЗ Колягин Ткачёва 7 класс (Алгебра)

1) xy^2-by^2-ax+ab+y^2-a;
2) ax^2-ay-bx^2+cy+by-cx^2;
3) a^2 x^2-bx^2+a^2 x-bx+a^2 y-by;
4) ax^2-bx^2+ay-by-ax+bx.
Выполнить действия:
1) (64x^2 y^2-1)/(x^2-4)•(x+2)^2/(x^2-4)•(x-2)^2/(8xy+1);
2) (ab-4b-2a+8)/(2a+8-ab-4b) :(2a-8-ab+4b)/(ab+4b-2a-8);
3) (x-6)/(x^2+6x+9)•(x^2+4x+4)/(x^2+2)(x-2) •(x^3-9x)/(x-6)(x+2) ;
4) (am^2-an^2)/(m^2+2mn+n^2 ) :(am^2-2amn+an^2)/(3m+3n).

1. $$xy^2-by^2-ax+ab+y^2-a=(xy^2-by^2+y^2)-(ax-ab+a)$$
   $$=y^2(x-b+1)-a(x-b+1)=(x-b+1)(y^2-a).$$

   Проверка:
   $$ (x-b+1)(y^2-a)=xy^2-by^2+y^2-ax+ab-a. $$

2. $$ax^2-ay-bx^2+cy+by-cx^2=(ax^2-bx^2-cx^2)-(ay-cy-by)$$
   $$=x^2(a-b-c)-y(a-c-b)=(a-b-c)(x^2-y).$$

   Проверка:
   $$ (a-b-c)(x^2-y)=ax^2-ay-bx^2+by-cx^2+cy. $$

3. $$a^2x^2-bx^2+a^2x-bx+a^2y-by=(a^2x^2+a^2x+a^2y)-(bx^2+bx+by)$$
   $$=a^2(x^2+x+y)-b(x^2+x+y)=(x^2+x+y)(a^2-b).$$

   Проверка:
   $$ (x^2+x+y)(a^2-b)=a^2x^2-bx^2+a^2x-bx+a^2y-by. $$

4. $$ax^2-bx^2+ay-by-ax+bx=(ax^2+ay-ax)-(bx^2+by-bx)$$
   $$=a(x^2+y-x)-b(x^2+y-x)=(x^2+y-x)(a-b).$$

   Проверка:
   $$ (x^2+y-x)(a-b)=ax^2-bx^2+ay-by-ax+bx. $$

$$\frac{64x^2y^2-1}{x^2-4}\cdot\frac{(x+2)^2}{x^2-4}\cdot\frac{(x-2)^2}{8xy+1}$$
$$=\frac{(8xy-1)(8xy+1)(x+2)^2(x-2)^2}{(x-2)(x+2)(x-2)(x+2)(8xy+1)}$$
$$=8xy-1.$$

$$\frac{ab-4b-2a+8}{2a+8-ab-4b}:\frac{2a-8-ab+4b}{ab+4b-2a-8}$$
$$=\frac{b(a-4)-2(a-4)}{2(a+4)-b(a+4)}:\frac{2(a-4)-b(a-4)}{b(a+4)-2(a+4)}$$
$$=\frac{(a-4)(b-2)}{(a+4)(2-b)}:\frac{(a-4)(2-b)}{(a+4)(b-2)}=1.$$

$$\frac{x-6}{x^2+6x+9}\cdot\frac{x^2+4x+4}{(x^2+2)(x-2)}\cdot\frac{x^3-9x}{(x-6)(x+2)}$$
$$=\frac{x-6}{(x+3)^2}\cdot\frac{(x+2)^2}{(x^2+2)(x-2)}\cdot\frac{x(x-3)(x+3)}{(x-6)(x+2)}$$
$$=\frac{x(x+2)(x-3)}{(x+3)(x^2+2)(x-2)}.$$

$$\frac{am^2-an^2}{m^2+2mn+n^2}:\frac{am^2-2amn+an^2}{3m+3n}$$
$$=\frac{a(m-n)(m+n)}{(m+n)^2}:\frac{a(m-n)^2}{3(m+n)}$$
$$=\frac{a(m-n)}{m+n}\cdot\frac{3(m+n)}{a(m-n)^2}=\frac{3}{m-n}.$$

Ответ

1) $$(x-b+1)(y^2-a)$$
2) $$(a-b-c)(x^2-y)$$
3) $$(x^2+x+y)(a^2-b)$$
4) $$(x^2+y-x)(a-b)$$
1) $$8xy-1$$
2) $$1$$
3) $$\frac{x(x+2)(x-3)}{(x+3)(x^2+2)(x-2)}$$
4) $$\frac{3}{m-n}$$