№ 4.33 Алгебра = № 4.33 Математика
Спростіть вираз:
1) $\frac{x-1}{x^2-x+1}+\frac{2-x}{x^3+1};$
2)$\ \frac{2m}{m-5}-\frac{5}{m+5}+\frac{2m^2}{25-m^2};$
3) $\frac{6}{m^2-6m}+\frac{m-12}{6m-36};$
4) $\frac{3}{2a+6}+\frac{a^2-a-3}{a^2-9}-1.$
Розв'язок:
1) $\frac{x-1}{x^2-x+1}+\frac{2-x}{x^3+1}=$
$= \frac{x-1}{x^2-x+1}+\frac{2-x}{(x+1)(x^2-x+1)}=$
$= \frac{\left(x-1\right)\left(x+1\right)+2-x}{\left(x+1\right)\left(x^2-x+1\right)}= $
$= \frac{x^2-1+2-x}{\left(x+1\right)\left(x^2-x+1\right)}=$
$= \frac{x^2-x+1}{\left(x+1\right)\left(x^2-x+1\right)}=\frac{1}{x+1};$
2)$\ \frac{2m}{m-5}-\frac{5}{m+5}+\frac{2m^2}{25-m^2}=$
$= \frac{2m}{m-5}-\frac{5}{m+5}-\frac{2m^2}{\left(m-5\right)\left(m+5\right)}=$
$= \frac{2m\left(m+5\right)-5\left(m-5\right)-2m^2}{\left(m-5\right)\left(m+5\right)}=$
$= \frac{2m^2+10m-5m+25-2m^2}{\left(m-5\right)\left(m+5\right)}=$
$= \frac{5m+25}{\left(m-5\right)\left(m+5\right)}=$
$= \frac{5\left(m+5\right)}{\left(m-5\right)\left(m+5\right)}=\frac{5}{m-5};$
3) $\frac{6}{m^2-6m}+\frac{m-12}{6m-36}=$
$= \frac{6}{m\left(m-6\right)}+\frac{m-12}{6\left(m-6\right)}=$
$= \frac{36+m(m-12)}{6m\left(m-6\right)}=$
$= \frac{36+m^2-12m}{6m\left(m-6\right)}= \frac{\left(m-6\right)^2}{6m\left(m-6\right)}=$
$= \frac{m-6}{6m};$
4) $\frac{3}{2a+6}+\frac{a^2-a-3}{a^2-9}-1=$
$= \frac{3}{2\left(a+3\right)}+\frac{a^2-a-3}{\left(a-3\right)\left(a+3\right)}-1=$
$= \frac{3(a-3)+2(a^2-a-3)-2(a^2-9)}{2\left(a+3\right)(a-3)}=$
$= \frac{3a-9+2a^2-2a-6-2a^2+18}{2\left(a+3\right)(a-3)}=$
$= \frac{a+3}{2\left(a+3\right)(a-3)}=$
$= \frac{1}{2\left(a-3\right)}=\frac{1}{2a-6}.$