Упр.44 Повторение ГДЗ Колмогоров 10-11 класс (Алгебра)

Задача

а) (m+n-4mn/(m+n)):(m/(m+n)-n/(n-m)-2mn/(m^2-n^2));
б) (a^3+b^3)/(a+b):(a^2-b^2)+2b/(a+b)-ab/(a^2-b^2);
в) (x/(x^2-4)-8/(x^2+2x))·(x^2-2x)/(4-x)+(x+8)/(x+2);
г) (1/(c^2+3c+2)+2c/(c^2+4c+3)+1/(c^2+5c+6))^2·((c-3)^2+12c)/2.

Подробный ответ

а)

$$ \left(m+n-\frac{4mn}{m+n}\right):\left(\frac{m}{m+n}-\frac{n}{n-m}-\frac{2mn}{m^2-n^2}\right) $$
$$ m+n-\frac{4mn}{m+n}=\frac{(m+n)^2-4mn}{m+n}=\frac{m^2-2mn+n^2}{m+n}=\frac{(m-n)^2}{m+n} $$
$$ \frac{m}{m+n}-\frac{n}{n-m}-\frac{2mn}{m^2-n^2} =\frac{m}{m+n}+\frac{n}{m-n}-\frac{2mn}{(m-n)(m+n)} $$
$$ =\frac{m(m-n)+n(m+n)-2mn}{(m-n)(m+n)} =\frac{m^2-2mn+n^2}{(m-n)(m+n)} =\frac{m-n}{m+n} $$
$$ \frac{(m-n)^2}{m+n}:\frac{m-n}{m+n}=m-n $$

б)

$$ \frac{a^3+b^3}{a+b}:(a^2-b^2)+\frac{2b}{a+b}-\frac{ab}{a^2-b^2} $$
$$ \frac{a^3+b^3}{a+b}=\frac{(a+b)(a^2-ab+b^2)}{a+b}=a^2-ab+b^2 $$
$$ a^2-b^2=(a-b)(a+b) $$
$$ \frac{a^2-ab+b^2}{(a-b)(a+b)}+\frac{2b}{a+b}-\frac{ab}{(a-b)(a+b)} $$
$$ =\frac{a^2-ab+b^2+2b(a-b)-ab}{(a-b)(a+b)} =\frac{a^2-b^2}{a^2-b^2}=1 $$

в)

$$ \left(\frac{x}{x^2-4}-\frac{8}{x^2+2x}\right)\cdot\frac{x^2-2x}{4-x}+\frac{x+8}{x+2} $$
$$ x^2-4=(x-2)(x+2), \qquad x^2+2x=x(x+2), \qquad 4-x=-(x-4) $$
$$ \frac{x}{(x-2)(x+2)}-\frac{8}{x(x+2)} =\frac{x^2-8x+16}{x(x-2)(x+2)} =\frac{(x-4)^2}{x(x-2)(x+2)} $$
$$ \frac{x^2-2x}{4-x}=\frac{x(x-2)}{4-x} $$
$$ \frac{(x-4)^2}{x(x-2)(x+2)}\cdot\frac{x(x-2)}{4-x} =\frac{x-4}{x+2} $$
$$ \frac{x-4}{x+2}+\frac{x+8}{x+2} =\frac{2x+4}{x+2}=2 $$

г)

$$ \left(\frac{1}{c^2+3c+2}+\frac{2c}{c^2+4c+3}+\frac{1}{c^2+5c+6}\right)^2\cdot\frac{(c-3)^2+12c}{2} $$
$$ c^2+3c+2=(c+1)(c+2), \quad c^2+4c+3=(c+1)(c+3), \quad c^2+5c+6=(c+2)(c+3) $$
$$ \frac{1}{(c+1)(c+2)}+\frac{2c}{(c+1)(c+3)}+\frac{1}{(c+2)(c+3)} =\frac{(c+3)+2c(c+2)+(c+1)}{(c+1)(c+2)(c+3)} $$
$$ =\frac{2(c+1)(c+2)}{(c+1)(c+2)(c+3)}=\frac{2}{c+3} $$
$$ \left(\frac{2}{c+3}\right)^2\cdot\frac{(c-3)^2+12c}{2} =\frac{4}{(c+3)^2}\cdot\frac{c^2+6c+9}{2} =\frac{4}{(c+3)^2}\cdot\frac{(c+3)^2}{2}=2 $$