Упр.503 ГДЗ Колягин Ткачёва 7 класс (Алгебра)

1) x^2 (x-3)-x(x-3)^2;
2) a^3 (2+a)+a^2 (2+a)^2;
3) 3m(n-m)^2-9m^2 (m-n);
4) 15p^2 (p+q)-5p(p+q)^2.
Выполнить действия:
1) ((c-d)/(c^2+dc)-c/(d^2+cd)) :(d^2/(c^3-cd^2 )+1/(c+d));
2) (2n/(k+2n)-(4n^2)/(k^2+4nk+4n^2 )) :(2n/(k^2-4n^2 )+1/(2n-k));
3) (2q/(2q+m)-(4q^2)/(4q^2+4mq+m^2 )) :(2q/(4q^2-m^2 )+1/(m-2q)).

1. $$x^2(x-3)-x(x-3)^2=(x-3)\bigl(x^2-x(x-3)\bigr)$$
   $$=(x-3)(x^2-x^2+3x)=3x(x-3).$$

2. $$a^3(2+a)+a^2(2+a)^2=a^2(2+a)\bigl(a+(2+a)\bigr)$$
   $$=a^2(2+a)(2a+2)=2a^2(a+1)(a+2).$$

3. $$3m(n-m)^2-9m^2(m-n)=3m(n-m)^2+9m^2(n-m)$$
   $$=3m(n-m)\bigl((n-m)+3m\bigr)=3m(n-m)(n+2m).$$

4. $$15p^2(p+q)-5p(p+q)^2=5p(p+q)\bigl(3p-(p+q)\bigr)$$
   $$=5p(p+q)(2p-q).$$

1)
$$\left(\frac{c-d}{c^2+dc}-\frac{c}{d^2+cd}\right):\left(\frac{d^2}{c^3-cd^2}+\frac{1}{c+d}\right)$$
$$=\left(\frac{c-d}{c(c+d)}-\frac{c}{d(c+d)}\right):\left(\frac{d^2}{c(c-d)(c+d)}+\frac{1}{c+d}\right)$$
$$=\frac{d(c-d)-c^2}{cd(c+d)}:\frac{d^2+c(c-d)}{c(c-d)(c+d)}$$
$$=\frac{-(c^2+d^2-cd)}{cd(c+d)}:\frac{c^2+d^2-cd}{c(c-d)(c+d)}$$
$$=-\frac{1}{d}\cdot(c-d)=\frac{d-c}{d}.$$

2)
$$\left(\frac{2n}{k+2n}-\frac{4n^2}{k^2+4nk+4n^2}\right):\left(\frac{2n}{k^2-4n^2}+\frac{1}{2n-k}\right)$$
$$=\left(\frac{2n}{k+2n}-\frac{4n^2}{(k+2n)^2}\right):\left(\frac{2n}{(k-2n)(k+2n)}-\frac{1}{k-2n}\right)$$
$$=\frac{2n(k+2n)-4n^2}{(k+2n)^2}:\frac{2n-(k+2n)}{(k-2n)(k+2n)}$$
$$=\frac{2nk}{(k+2n)^2}:\frac{-k}{(k-2n)(k+2n)}$$
$$=\frac{2nk}{(k+2n)^2}\cdot\frac{(k-2n)(k+2n)}{-k}$$
$$=\frac{4n^2-2nk}{k+2n}.$$

3)
$$\left(\frac{2q}{2q+m}-\frac{4q^2}{4q^2+4mq+m^2}\right):\left(\frac{2q}{4q^2-m^2}+\frac{1}{m-2q}\right)$$
$$=\left(\frac{2q}{2q+m}-\frac{4q^2}{(2q+m)^2}\right):\left(\frac{2q}{(2q-m)(2q+m)}-\frac{1}{2q-m}\right)$$
$$=\frac{2q(2q+m)-4q^2}{(2q+m)^2}:\frac{2q-(2q+m)}{(2q-m)(2q+m)}$$
$$=\frac{2qm}{(2q+m)^2}:\frac{-m}{(2q-m)(2q+m)}$$
$$=\frac{2qm}{(2q+m)^2}\cdot\frac{(2q-m)(2q+m)}{-m}$$
$$=\frac{4q^2-2qm}{2q+m}.$$

Ответ

1) $$3x(x-3)$$; 2) $$2a^2(a+1)(a+2)$$; 3) $$3m(n-m)(n+2m)$$; 4) $$5p(p+q)(2p-q)$$; 5) $$\frac{d-c}{d}$$; 6) $$\frac{4n^2-2nk}{k+2n}$$; 7) $$\frac{4q^2-2qm}{2q+m}$$.