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Bài 1:
a) Ta có: \(P=1+\dfrac{3}{x^2+5x+6}:\left(\dfrac{8x^2}{4x^3-8x^2}-\dfrac{3x}{3x^2-12}-\dfrac{1}{x+2}\right)\)
\(=1+\dfrac{3}{\left(x+2\right)\left(x+3\right)}:\left(\dfrac{8x^2}{4x^2\left(x-2\right)}-\dfrac{3x}{3\left(x-2\right)\left(x+2\right)}-\dfrac{1}{x+2}\right)\)
\(=1+\dfrac{3}{\left(x+2\right)\left(x+3\right)}:\left(\dfrac{4}{x-2}-\dfrac{x}{\left(x-2\right)\left(x+2\right)}-\dfrac{1}{x+2}\right)\)
\(=1+\dfrac{3}{\left(x+2\right)\left(x+3\right)}:\dfrac{4\left(x+2\right)-x-\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}\)
\(=1+\dfrac{3}{\left(x+2\right)\left(x+3\right)}\cdot\dfrac{\left(x-2\right)\left(x+2\right)}{4x+8-x-x+2}\)
\(=1+3\cdot\dfrac{\left(x-2\right)}{\left(x+3\right)\left(2x+10\right)}\)
\(=1+\dfrac{3\left(x-2\right)}{\left(x+3\right)\left(2x+10\right)}\)
\(=\dfrac{\left(x+3\right)\left(2x+10\right)+3\left(x-2\right)}{\left(x+3\right)\left(2x+10\right)}\)
\(=\dfrac{2x^2+10x+6x+30+3x-6}{\left(x+3\right)\left(2x+10\right)}\)
\(=\dfrac{2x^2+19x-6}{\left(x+3\right)\left(2x+10\right)}\)
a) \(ĐKXĐ:\hept{\begin{cases}x\ne0\\x\ne2\end{cases}}\)
\(Q=\left(\frac{2x-x^2}{2x^2+8}-\frac{2x^2}{x^3-2x^2+4x-8}\right).\left(\frac{2}{x^2}+\frac{1-x}{x}\right)\)
\(\Leftrightarrow Q=\left(\frac{x\left(2-x\right)}{2\left(x^2+4\right)}-\frac{2x^2}{\left(x-2\right)\left(x^2+4\right)}\right).\frac{2+x\left(1-x\right)}{x^2}\)
\(\Leftrightarrow Q=\frac{-x\left(x-2\right)^2-4x^2}{2\left(x-2\right)\left(x^2+4\right)}.\frac{2+x-x^2}{x^2}\)
\(\Leftrightarrow Q=\frac{x\left(x^2-4x+4\right)-4x^2}{2\left(x-2\right)\left(x^2+4\right)}.\frac{\left(x-2\right)\left(x+1\right)}{x^2}\)
\(\Leftrightarrow Q=\frac{x\left(x^2+4\right)}{2\left(x^2+4\right)}.\frac{x+1}{x^2}\)
\(\Leftrightarrow Q=\frac{x+1}{2x}\)
b) Để \(Q\inℤ\)
\(\Leftrightarrow x+1⋮2x\)
\(\Leftrightarrow2\left(x+1\right)⋮2x\)
\(\Leftrightarrow2x+2⋮2x\)
\(\Leftrightarrow2⋮2x\)
\(\Leftrightarrow2x\inƯ\left(2\right)\)
\(\Leftrightarrow2x\in\left\{\pm1;\pm2\right\}\)
\(\Leftrightarrow x\in\left\{\pm\frac{1}{2};\pm1\right\}\)
Mà \(x\inℤ\)
Vậy để \(Q\inℤ\Leftrightarrow x\in\left\{1;-1\right\}\)
