14: \(\frac{2x+x^2}{x^3-1}-\frac{1}{x-1}\)
\(=\frac{x^2+2x}{\left(x-1\right)\left(x^2+x+1\right)}-\frac{1}{x-1}\)
\(=\frac{x^2+2x-x^2-x-1}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{x-1}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{1}{x^2+x+1}\)
\(1-\frac{x+2}{x^2+x+1}\)
\(=\frac{x^2+x+1-x-2}{x^2+x+1}=\frac{x^2-1}{x^2+x+1}=\frac{\left(x-1\right)\left(x+1\right)}{x^2+x+1}\)
\(T=\left(\frac{2x+x^2}{x^3-1}-\frac{1}{x-1}\right):\left(1-\frac{x+2}{x^2+x+1}\right)\)
\(=\frac{1}{x^2+x+1}:\frac{\left(x-1\right)\left(x+1\right)}{x^2+x+1}\)
\(=\frac{1}{x^2+x+1}\cdot\frac{x^2+x+1}{x^2-1}=\frac{1}{x^2-1}\)
13: \(\frac{x^3-3x}{x^2-9}-1\)
\(=\frac{x^3-3x-x^2+9}{x^2-9}\)
\(\frac{9-x^2}{\left(x+3\right)\left(x-2\right)}+\frac{x-3}{x-2}-\frac{x+2}{x+3}\)
\(=\frac{\left(3-x\right)\left(3+x\right)}{\left(3+x\right)\left(x+2\right)}+\frac{x-3}{x-2}-\frac{x+2}{x+3}=\frac{x+3}{x+2}+\frac{x-3}{x-2}-\frac{x+2}{x+3}=\frac{x-3}{x-2}\)
\(S=\left(\frac{x^3-3x}{x^2-9}-1\right):\left\lbrack\frac{9-x^2}{\left(x+3\right)\left(x-2\right)}+\frac{x-3}{x-2}-\frac{x+2}{x+3}\right\rbrack\)
\(=\frac{x^3-x^2-3x+9}{x^2-9}:\frac{x-3}{x-2}=\frac{x^3-x^2-3x+9}{x^2-9}\cdot\frac{x-2}{x-3}=\frac{\left(x^3-x^2-3x+9\right)\left(x-2\right)}{\left(x-3\right)^2\cdot\left(x+3\right)}\)
12: \(\frac{x^2+2}{x^3-1}+\frac{x}{x^2+x+1}+\frac{1}{1-x}\)
\(=\frac{x^2+2}{\left(x-1\right)\left(x^2+x+1\right)}+\frac{x}{x^2+x+1}-\frac{1}{x-1}\)
\(=\frac{x^2+2+x\left(x-1\right)-x^2-x-1}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{-x+1+x^2-x}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\frac{x^2-2x+1}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{\left(x-1\right)^2}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{x-1}{x^2+x+1}\)
\(Q=\left(\frac{x^2+2}{x^3-1}+\frac{x}{x^2+x+1}+\frac{1}{1-x}\right):\frac{x-1}{2}\)
\(=\frac{x-1}{x^2+x+1}\cdot\frac{2}{x-1}=\frac{2}{x^2+x+1}\)
11: \(\frac{x}{x+2}-\frac{x^3-8}{x^3+8}\cdot\frac{x^2-2x+4}{x^2-4}\)
\(=\frac{x}{x+2}-\frac{\left(x-2\right)\left(x^2+2x+4\right)}{\left(x+2\right)\left(x^2-2x+4\right)}\cdot\frac{x^2-2x+4}{\left(x-2\right)\left(x+2\right)}\)
\(=\frac{x}{x+2}-\frac{x^2+2x+4}{\left(x+2\right)^2}=\frac{x\left(x+2\right)-x^2-2x-4}{\left(x+2\right)^2}=\frac{-4}{\left(x+2\right)^2}\)
\(P=\left(\frac{x}{x+2}-\frac{x^3-8}{x^3+8}\cdot\frac{x^2-2x+4}{x^2-4}\right):\frac{4}{x+2}\)
\(=-\frac{4}{\left(x+2\right)^2}\cdot\frac{x+2}{4}=\frac{-1}{\left(x+2\right)}\)
10: \(\frac{\left(a-b\right)^2+4ab}{a+b}\)
\(=\frac{a^2-2ab+b^2+4ab}{a+b}=\frac{a^2+2ab+b^2}{a+b}=\frac{\left(a+b\right)^2}{a+b}\)
=a+b
\(\frac{a^2b-ab^2}{ab}=ab\cdot\frac{\left(a-b\right)}{ab}=a-b\)
\(N=\frac{\left(a-b\right)^2+4ab}{a+b}-\frac{a^2b-b^2a}{ab}\)
=a+b-(a-b)
=a+b-a+b
=2b
9: \(\frac{5x+1}{x^3-1}-\frac{1-2x}{x^2+x+1}-\frac{2}{1-x}\)
\(=\frac{5x+1}{\left(x-1\right)\cdot\left(x^2+x+1\right)}+\frac{2x-1}{x^2+x+1}+\frac{2}{x-1}\)
\(=\frac{5x+1+\left(2x-1\right)\left(x-1\right)+2\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\frac{5x+1+2x^2-3x+1+2x^2+2x+2}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{4x^2+4x+4}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\frac{4\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{4}{x-1}\)
\(M=\left(\frac{5x+1}{x^3-1}-\frac{1-2x}{x^2+x+1}-\frac{2}{1-x}\right):\frac{2x}{x^2-1}\)
\(=\frac{4}{x-1}\cdot\frac{x^2-1}{2x}=\frac{4}{2x}\cdot\frac{\left(x-1\right)\left(x+1\right)}{x-1}=\frac{2}{x}\cdot\left(x+1\right)=\frac{2x+2}{x}\)
8: \(\frac{x+x^3}{1-x^2}-\frac{x-x^3}{1+x^2}\)
\(=\frac{\left(x+x^3\right)\left(1+x^2\right)-\left(x-x^3\right)\left(1-x^2\right)}{\left(1-x^2\right)\left(1+x^2\right)}\)
\(=\frac{x+x^3+x^3+x^5-\left(x-x^3-x^3+x^5\right)}{\left(1-x^2\right)\left(1+x^2\right)}=\frac{4x^3}{\left(1-x\right)\left(1+x\right)\left(1+x^2\right)}\)
\(\frac{1+x}{1-x}-\frac{1-x}{1+x}\)
\(=\frac{\left(1+x\right)^2-\left(1-x\right)^2}{\left(1-x\right)\left(1+x\right)}\)
\(=\frac{\left(1+x-1+x\right)\left(1+x+1-x\right)}{\left(1-x\right)\left(1+x\right)}=\frac{2x\cdot2}{\left(1-x\right)\left(1+x\right)}=\frac{4x}{\left(1-x\right)\left(1+x\right)}\)
\(I=\left(\frac{x+x^3}{1-x^2}-\frac{x-x^3}{1+x^2}\right):\left(\frac{1+x}{1-x}-\frac{1-x}{1+x}\right)\)
\(=\frac{4x^3}{\left(1-x\right)\left(1+x\right)\left(1+x^2\right)}:\frac{4x}{\left(1-x\right)\left(1+x\right)}\)
\(=\frac{4x^3}{\left(1-x\right)\left(1+x\right)\left(1+x^2\right)}\cdot\frac{\left(1-x\right)\left(1+x\right)}{4x}=\frac{x^2}{x^2+1}\)
7: \(\frac{2x+1}{2x+4}-\frac{x}{3x-6}-\frac{2x^2}{3x^2-12}\)
\(=\frac{2x+1}{2\left(x+2\right)}-\frac{x}{3\left(x-2\right)}-\frac{2x^2}{3\left(x-2\right)\left(x+2\right)}\)
\(=\frac{\left(2x+1\right)\left(3x-6\right)-2x\left(x+2\right)-4x^2}{6\left(x-2\right)\left(x+2\right)}\)
\(=\frac{6x_{}^2-12x+3x-6-2x^2-4x-4x^2}{6\left(x-2\right)\left(x+2\right)}=\frac{-13x-6}{6\left(x-2\right)\left(x+2\right)}\)
\(H=\left(\frac{2x+1}{2x+4}-\frac{x}{3x-6}-\frac{2x^2}{3x^2-12}\right)\cdot\frac{24-12x}{6+13x}\)
\(=\frac{-\left(13x+6\right)}{6\left(x-2\right)\left(x+2\right)}\cdot\frac{12\left(2-x\right)}{13x+6}\)
\(=\frac{-12\left(2-x\right)}{6\left(x-2\right)\left(x+2\right)}=\frac{12\left(x-2\right)}{6\left(x-2\right)\left(x+2\right)}=\frac{2}{x+2}\)
6: \(\frac{x+2}{3x}+\frac{2}{x+1}-3\)
\(=\frac{\left(x+2\right)\left(x+1\right)+2\cdot3x-3\cdot3x\left(x+1\right)}{3x\left(x+1\right)}\)
\(=\frac{x^2+3x+2+6x-9x^2-9x}{3x\left(x+1\right)}=\frac{-8x^2+2}{3x\left(x+1\right)}\)
\(G=\left(\frac{x+2}{3x}+\frac{2}{x+1}-3\right):\frac{2-4x}{x+1}+\frac{x^2-3x-1}{3x}\)
\(=\frac{-8x^2+2}{3x\left(x+1\right)}\cdot\frac{x+1}{-2\left(2x-1\right)}+\frac{x^2-3x+1}{3x}\)
\(=\frac{-2\left(2x-1\right)\left(2x+1\right)}{3x}\cdot\frac{1}{-2\left(2x-1\right)}+\frac{x^2-3x+1}{3x}\)
\(=\frac{2x+1}{3x}+\frac{x^2-3x+1}{3x}=\frac{x^2-x+2}{3x}\)
5: \(\frac{x}{x^2-36}-\frac{x-6}{x^2+6x}\)
\(=\frac{x}{\left(x-6\right)\left(x+6\right)}-\frac{x-6}{x\left(x+6\right)}\)
\(=\frac{x^2-\left(x-6\right)^2}{x\left(x-6\right)\left(x+6\right)}=\frac{x^2-x^2+12x-36}{x\left(x-6\right)\left(x+6\right)}=\frac{12\left(x-3\right)}{x\left(x-6\right)\left(x+6\right)}\)
\(E=\left(\frac{x}{x^2-36}-\frac{x-6}{x^2+6x}\right):\frac{2x-6}{x^2+6x}+\frac{x}{6-x}\)
\(=\frac{12\left(x-3\right)}{x\left(x-6\right)\left(x+6\right)}\cdot\frac{x\left(x+6\right)}{2\left(x-3\right)}-\frac{x}{x-6}=\frac{6}{x-6}-\frac{x}{x-6}=\frac{6-x}{x-6}\)
=-1
4: \(\frac{y+3}{y^2-3y}-\frac{y}{y^2-9}\)
\(=\frac{y+3}{y\left(y-3\right)}-\frac{y}{\left(y-3\right)\left(y+3\right)}\)
\(=\frac{\left(y+3\right)^2-y^2}{y\left(y-3\right)\left(y+3\right)}=\frac{3\left(2y+3\right)}{y\left(y-3\right)\left(y+3\right)}\)
\(D=\frac{y}{3-y}+\frac{y^2+3y}{2y+3}\cdot\left(\frac{y+3}{y^2-3y}-\frac{y}{y^2-9}\right)\)
\(=\frac{-y}{y-3}+\frac{y\left(y+3\right)}{2y+3}\cdot\frac{3\left(2y+3\right)}{y\left(y-3\right)\left(y+3\right)}\)
\(=\frac{-y}{y-3}+\frac{3}{y-3}=\frac{-y+3}{y-3}=-1\)
3: \(\frac{x}{x-1}-\frac{x+1}{x}\)
\(=\frac{x^2-\left(x+1\right)\left(x-1\right)}{x\left(x-1\right)}\)
\(=\frac{x^2-\left(x^2-1\right)}{x\left(x-1\right)}=\frac{1}{x\left(x-1\right)}\)
\(\frac{x}{x+1}-\frac{x-1}{x}\)
\(=\frac{x^2-\left(x-1\right)\left(x+1\right)}{x\left(x+1\right)}=\frac{x^2-\left(x^2-1\right)}{x\left(x+1\right)}=\frac{1}{x\left(x+1\right)}\)
\(C=\left(\frac{x}{x-1}-\frac{x+1}{x}\right):\left(\frac{x}{x+1}-\frac{x-1}{x}\right)\)
\(=\frac{1}{x\left(x-1\right)}:\frac{1}{x\left(x+1\right)}=\frac{x\left(x+1\right)}{x\left(x-1\right)}=\frac{x+1}{x-1}\)
2: \(B=\frac{x^2-1}{x+10}\cdot\frac{2x}{x+2}+\frac{x^2-1}{x+10}\cdot\frac{10-x}{x+2}\)
\(=\frac{x^2-1}{x+10}\cdot\frac{2x+10-x}{x+2}=\frac{x^2-1}{x+10}\cdot\frac{x+10}{x+2}=\frac{x^2-1}{x+2}\)
1: \(A=\left(\frac{6x+1}{x^2-6x}+\frac{6x-1}{x^2+6x}\right)\cdot\frac{x^2-36}{x^2+1}\)
\(=\left(\frac{6x+1}{x\left(x-6\right)}+\frac{6x-1}{x\left(x+6\right)}\right)\cdot\frac{\left(x-6\right)\left(x+6\right)}{x^2+1}\)
\(=\frac{\left(6x+1\right)\left(x+6\right)+\left(6x-1\right)\left(x-6\right)}{x\left(x-6\right)\left(x+6\right)}\cdot\frac{\left(x-6\right)\left(x+6\right)}{x^2+1}\)
\(=\frac{6x^2+36x+x+6+6x^2-36x-x+6}{x}\cdot\frac{1}{x^2+1}=\frac{12\left(x^2+1\right)}{x\left(x^2+1\right)}=\frac{12}{x}\)
