Упр.204(стар учебник) ГДЗ Дорофеев Суворова 8 класс (Алгебра)
а) (x^2-y^2+ax+ay)/(a^2+xy+ax-y^2 );
б) (m^3+m^2-m-1)/(m^3-m^2-m+1);
в) (y^4-2y^2+1)/(y^3-y^2-y+1);
г) (p^3+pq^2-2p^2 q)/(p^3-pq^2 );
д) (x^4-y^4)/(x^4+2x^3 y+2x^2 y^2+2xy^3+y^4 );
е) (x^4-y^4)/(x^3+xy^2-x^2 y-y^3 ).
а)
$$\frac{x^2-y^2+ax+ay}{a^2+xy+ax-y^2}= \frac{(x-y)(x+y)+a(x+y)}{(a-y)(a+y)+x(a+y)}= \frac{(x+y)(x-y+a)}{(a+y)(a-y+x)}$$
$$=\frac{x+y}{a+y}.$$б)
$$\frac{m^3+m^2-m-1}{m^3-m^2-m+1}= \frac{m^2(m+1)-(m+1)}{m^2(m-1)-(m-1)}= \frac{(m+1)(m^2-1)}{(m-1)(m^2-1)}= \frac{m+1}{m-1}.$$
в)
$$\frac{y^4-2y^2+1}{y^3-y^2-y+1}= \frac{(y^2-1)^2}{y^2(y-1)-(y-1)}= \frac{(y^2-1)^2}{(y-1)(y^2-1)}$$
$$=\frac{y^2-1}{y-1}= \frac{(y-1)(y+1)}{y-1}=y+1.$$г)
$$\frac{p^3+pq^2-2p^2q}{p^3-pq^2}= \frac{p(p^2-2pq+q^2)}{p(p^2-q^2)}= \frac{(p-q)^2}{(p-q)(p+q)}= \frac{p-q}{p+q}.$$
д)
$$\frac{x^4-y^4}{x^4+2x^3y+2x^2y^2+2xy^3+y^4}= \frac{(x^2-y^2)(x^2+y^2)}{(x^2+y^2)^2+2xy(x^2+y^2)}$$
$$=\frac{(x^2-y^2)(x^2+y^2)}{(x^2+y^2)(x^2+y^2+2xy)}= \frac{x^2-y^2}{(x+y)^2}= \frac{(x-y)(x+y)}{(x+y)^2}= \frac{x-y}{x+y}.$$е)
$$\frac{x^4-y^4}{x^3+xy^2-x^2y-y^3}= \frac{(x^2-y^2)(x^2+y^2)}{x^3-y^3-xy(x-y)}$$
$$= \frac{(x^2-y^2)(x^2+y^2)}{(x-y)(x^2+xy+y^2)-xy(x-y)}= \frac{(x^2-y^2)(x^2+y^2)}{(x-y)(x^2+xy+y^2-xy)}$$
$$= \frac{(x^2-y^2)(x^2+y^2)}{(x-y)(x^2+y^2)}= \frac{x^2-y^2}{x-y}= \frac{(x-y)(x+y)}{x-y}=x+y.$$
Ответ
а) $$\frac{x+y}{a+y}$$; б) $$\frac{m+1}{m-1}$$; в) $$y+1$$; г) $$\frac{p-q}{p+q}$$; д) $$\frac{x-y}{x+y}$$; е) $$x+y$$.
