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\(\frac{x^2+3x+9}{2x+10}.\frac{x+5}{x^3-27}\)
\(=\frac{x^2+3x+9}{2\left(x+5\right)}.\frac{x+5}{\left(x-3\right)\left(x^2+3x+9\right)}\)
\(=\frac{\left(x+5\right)\left(x^2+3x+9\right)}{2\left(x+5\right)\left(x-3\right)\left(x^2+3x+9\right)}\)
\(=\frac{1}{2\left(x-3\right)}\)
\(\left(\frac{6x+1}{x^2-6x}+\frac{6x-1}{x^2+6x}\right)\left(\frac{x^2-36}{x^2+1}\right)\)
\(=\left[\frac{6x+1}{x\left(x-6\right)}+\frac{6x-1}{x\left(x+6\right)}\right]\left[\frac{\left(x-6\right)\left(x+6\right)}{x^2+1}\right]\)
\(=\frac{\left(6x+1\right)\left(x+6\right)+\left(6x-1\right)\left(x-6\right)}{x\left(x-6\right)\left(x+6\right)}.\frac{\left(x-6\right)\left(x+6\right)}{x^2+1}\)
\(=\frac{6x^2+36x+x+6+6x^2-36x-x+6}{x\left(x-6\right)\left(x+6\right)}.\frac{\left(x-6\right)\left(x+6\right)}{x^2+1}\)
\(=\frac{12x^2+12}{x\left(x-6\right)\left(x+6\right)}.\frac{\left(x-6\right)\left(x+6\right)}{x^2+1}\)
\(=\frac{12\left(x^2+1\right).\left(x-6\right)\left(x+6\right)}{x\left(x-6\right)\left(x+6\right)\left(x^2+1\right)}\)
\(=\frac{12}{x}\)
a) 1/x(x + 1) + 1/(x + 1)(x + 2) + 1/(x + 2)(x + 3) + 1/(x + 3)(x + 4)
( 1/x - 1/x+1) + (1/x+1 - 1/x+2) + (1/x+2 - 1/ x+3) + 1/(x+3 - 1/x+4)
(1/x +1/x+4) - ( 1/x+2 - 1/x+2) - ( 1/x+3 - 1/x+3)
1/x +1/x+4
2x+4/x(x+4)
a: \(=\dfrac{x^2+3x+9}{2\left(x+5\right)}\cdot\dfrac{\left(x+5\right)}{\left(x-3\right)\left(x^2+3x+9\right)}=\dfrac{1}{2\left(x-3\right)}\)
b: \(=\dfrac{\left(6x+1\right)\left(x+6\right)+\left(6x-1\right)\left(x-6\right)}{x\left(x-6\right)\left(x+6\right)}\cdot\dfrac{\left(x-6\right)\left(x+6\right)}{x^2+1}\)
\(=\dfrac{6x^2+37x+6+6x^2-37x+6}{x}\cdot\dfrac{1}{x^2+1}=\dfrac{12}{x}\)
a: \(=\dfrac{x^3-x^2+x-1}{\left(x-2\right)\left(x+2\right)}-\dfrac{x-2}{\left(x+2\right)\left(x+1\right)}-\dfrac{3x}{\left(x-2\right)\left(x+1\right)}+\dfrac{2x+5}{\left(x+1\right)\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{\left(x-1\right)\left(x^2+1\right)\left(x+1\right)-x^2+4x-4-3x^2-6x+2x+5}{\left(x+2\right)\left(x+1\right)\left(x-2\right)}\)
\(=\dfrac{x^4-1-4x^2+1}{\left(x+2\right)\left(x-2\right)\left(x+1\right)}=\dfrac{x^2\left(x-2\right)\left(x+2\right)}{\left(x+2\right)\left(x-2\right)\left(x+1\right)}\)
=x^2/x+1
b: Sửa đề: \(\dfrac{19x^2-30x+9}{2x^3+54}-\dfrac{x-3}{2x^2+6x}-\dfrac{3x^2}{2x^2-6x+18}\) \(=\dfrac{19x^2-30x+9}{2\left(x+3\right)\left(x^2-3x+9\right)}-\dfrac{x-3}{2x\left(x+3\right)}-\dfrac{3x^2}{2\left(x^2-3x+9\right)}\)
\(=\dfrac{19x^3-30x^2+9x-\left(x-3\right)\left(x^2-3x+9\right)-3x^3\left(x+3\right)}{2x\left(x+3\right)\left(x^2-3x+9\right)}\)
\(=\dfrac{19x^3-30x^2+9x-3x^4-9x^3-\left(x^3-3x^2+9x-3x^2+9x-27\right)}{2x\left(x+3\right)\left(x^2-3x+9\right)}\)
\(=\dfrac{-3x^4+10x^3-30x^2+9x-x^3+6x^2-18x+27}{2x\left(x+3\right)\left(x^2-3x+9\right)}\)
\(=\dfrac{-3x^4+10x^3-24x^2-9x+27}{2x\left(x+3\right)\left(x^2-3x+9\right)}\)
\(=\dfrac{-3\left(x-1\right)}{\left(1+x\right)^2}\cdot\dfrac{x+1}{6\left(x-1\right)\left(x+1\right)}\left(x\ne\pm1\right)\\ =\dfrac{-1}{2\left(x+1\right)^2}\)