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\(\left(\frac{1}{x+1}-\frac{3}{x^3+1}+\frac{3}{x^2-x+1}\right):\frac{3x^2-3x+3}{\left(x+1\right)\left(x+2\right)}-\frac{2x-2}{x^2+2x}\left(x\ne-1;x\ne0;x\ne-2\right)\)
\(=\left(\frac{1}{x+1}-\frac{3}{\left(x+1\right)\left(x^2-x+1\right)}+\frac{3}{x^2-x+1}\right):\frac{3x^3-3x+3}{\left(x+1\right)\left(x+2\right)}-\frac{2\left(x-1\right)}{x\left(x+2\right)}\)
\(=\left(\frac{x^2-x+1}{\left(x+1\right)\left(x^2-x+1\right)}-\frac{3}{\left(x+1\right)\left(x^2-x+1\right)}+\frac{3x+3}{\left(x+1\right)\left(x^2-x+1\right)}\right)\)\(:\frac{3x^2-3x+3}{\left(x+1\right)\left(x+2\right)}-\frac{2\left(x-1\right)}{x\left(x+2\right)}\)
\(=\frac{x^2-x+1-3+3x+3}{\left(x+1\right)\left(x^2-x+1\right)}:\frac{3x^2-3x+3}{\left(x+1\right)\left(x+2\right)}-\frac{2\left(x-1\right)}{x\left(x+2\right)}\)
\(=\frac{x^2+2x+1}{\left(x+1\right)\left(x^2-x+1\right)}:\frac{3\left(x^2-x+1\right)}{\left(x+1\right)\left(x+1\right)}-\frac{2\left(x-1\right)}{x\left(x+2\right)}\)
\(=\frac{\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x^2-x+1\right)}\cdot\frac{\left(x+1\right)\left(x+2\right)}{3\left(x^2-x+1\right)}-\frac{2\left(x-1\right)}{x\left(x+2\right)}\)
\(=\frac{\left(x+2\right)^2\left(x+1\right)}{3\left(x^2-x+1\right)^2}-\frac{2\left(x-1\right)}{x\left(x+2\right)}\)
\(\left(\frac{1}{x}+1-\frac{3}{x^3+1}-\frac{3}{x^2-x+1}\right)\cdot\frac{3x^2-3x+3}{\left(x+1\right).\left(x+2\right)}-\frac{2x-2}{x^2+2x}\)
\(=\left(\frac{x+1}{x}-\frac{3}{\left(x+1\right).\left(x^2-x+1\right)}+\frac{3.\left(x+1\right)}{\left(x+1\right).\left(x^2-x+1\right)}\right)\cdot\frac{3.\left(x^2-x+1\right)}{\left(x+1\right).\left(x+2\right)}-\frac{2.\left(x-1\right)}{x.\left(x+2\right)}\)
\(=\left[\frac{\left(x+1\right)^2.\left(x^2-x+1\right)-3x+3x^2+3x}{x.\left(x+1\right).\left(x^2-x+1\right)}\right]\cdot\frac{3.\left(x^2-x+1\right)}{\left(x+1\right).\left(x+2\right)}-\frac{2.\left(x-1\right)}{x.\left(x+2\right)}\)
\(=\left[\frac{x^4+x^3+x+1+3x^2}{x.\left(x+1\right).\left(x^2-x+1\right)}\right]\cdot\frac{3.\left(x^2-x+1\right)}{\left(x+1\right).\left(x+2\right)}-\frac{2.\left(x-1\right)}{x.\left(x+2\right)}\)
\(=\frac{3x^4+3x^3+3x+3+9x^2}{x.\left(x+1\right)^2.\left(x+2\right)}-\frac{2.\left(x-1\right)}{x.\left(x+2\right)}=\frac{3x^4+3x^3+3x+3+9x^2}{x.\left(x+1\right)^2.\left(x+2\right)}-\frac{2x^3+2x^2-2x-2}{x.\left(x+1\right)^2.\left(x+2\right)}\)
\(=\frac{3x^4+x^3+7x^2+5x+5}{x.\left(x+1\right)^2.\left(x+2\right)}\)
a) Ta có :A = \(\left(\frac{\left(x-1\right)^2}{3x+\left(x-1\right)^2}-\frac{1-2x^2+4x}{x^3-1}+\frac{1}{x-1}\right):\frac{x^2+x}{x^3+x}\)
ĐK: \(\hept{\begin{cases}x\ne0\\x\ne1\end{cases}}\)
A = \(\left(\frac{\left(x-1\right)^2}{x^2+x+1}-\frac{1-2x^2+4x}{\left(x-1\right)\left(x^2+x+1\right)}+\frac{1}{x-1}\right):\frac{x\left(x+1\right)}{x\left(x^2+1\right)}\)
= \(\frac{\left(x-1\right)^3-1+2x^2-4x+x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}.\frac{x^2+1}{x+1}\)
= \(\frac{x^3-3x^2+3x-1+3x^2-3x}{\left(x-1\right)\left(x^2+x+1\right)}.\frac{x^2+1}{x+1}\)
= \(\frac{x^3-1}{\left(x-1\right)\left(x^2+x+1\right)}.\frac{x^2+1}{x+1}=1.\frac{x^2+1}{x+1}=\frac{x^2+1}{x+1}\)
b) Để A > - 1 <=> \(\frac{x^2+1}{x+1}>-1\)
<=> \(\frac{x^2+1}{x+1}+1>0\)
<=> \(\frac{x^2+x+2}{x+1}>0\)
Vì x2 + x + 2 >0 \(\forall x\)
=> A > 0 <=> x + 1 > 0 <=> x > -1
\(ĐK:x\ne\pm1;x\ne0;x\ne3\)
Với \(x\ne\pm1;x\ne0;x\ne3\)thì\(M=\frac{x^3+2x^2-x-2}{x^3-2x^2-3x}\left[\frac{\left(x+2\right)^2-x^2}{4x^2-4}-\frac{3}{x^2-x}\right]=\frac{x^2\left(x+2\right)-\left(x+2\right)}{\left(x^3-x\right)-\left(2x^2+2x\right)}\left[\frac{x^2+4x+4-x^2}{4x^2-4}-\frac{3}{x\left(x-1\right)}\right]\)\(=\frac{\left(x-1\right)\left(x+1\right)\left(x+2\right)}{x\left(x+1\right)\left(x-1\right)-2x\left(x+1\right)}\left[\frac{4\left(x+1\right)}{4\left(x+1\right)\left(x-1\right)}-\frac{3}{x\left(x-1\right)}\right]=\frac{\left(x-1\right)\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x^2-3x\right)}\left[\frac{1}{x-1}-\frac{3}{x\left(x-1\right)}\right]\)\(=\frac{\left(x-1\right)\left(x+2\right)}{x\left(x-3\right)}.\frac{x-3}{x\left(x-1\right)}=\frac{x+2}{x^2}\)
M = 3 \(\Leftrightarrow\frac{x+2}{x^2}=3\Leftrightarrow3x^2-x-2=0\Leftrightarrow\left(x-1\right)\left(3x+2\right)=0\Leftrightarrow\orbr{\begin{cases}x=1\\x=\frac{-2}{3}\end{cases}}\)
Mà \(x\ne1\)(theo điều kiện) nên x =-2/3
=[x(x-2)/2(x2+4)-2x2/(4+x2)(2-x)][x(x-2)(x+1)/x3]
={[x(x-2)(2-x)-4x2 ]/2(2-x)(4+x2)} .[x(x-2)(x+1)/x3 ]
=[-x(x2+4)/2(2-x)(4+x2)].[x(x-2)(x+1)/x3 ]
=-x.x(x-2)(x+1)/2(2-x)x3
=(x+1)/2x
a) A \(=\)\(\frac{\left(2x^2+2x\right)\left(x-2\right)^2}{\left(x^3-4x\right)\left(x+1\right)}\)\(=\)\(\frac{2x\left(x+1\right)\left(x-2\right)^2}{x\left(x-2\right)\left(x+2\right)\left(x+1\right)}\)
\(=\)\(\frac{2\left(x-2\right)}{x+2}\)\(=\)\(\frac{2x-4}{x+2}\)
Tại x = \(\frac{1}{2}\)thì:
A = \(\frac{2.\frac{1}{2}-4}{\frac{1}{2}+2}\)\(=\)\(\frac{-3}{\frac{5}{2}}\)\(=\)\(\frac{-6}{5}\)
\(P=\left(\frac{x+3}{3-x}-\frac{3-x}{x+3}+\frac{4x^2}{x^2-9}\right):\left(\frac{2x+1}{x+3}+1\right)\)\(\left(ĐKXĐ:x\ne\pm3\right)\).
\(P=\left(\frac{x+3}{3-x}-\frac{3-x}{x+3}-\frac{4x^2}{9-x^2}\right):\left(\frac{2x+1}{x+3}+\frac{x+3}{x+3}\right)\).
\(P=\left[\frac{\left(x+3\right)^2}{\left(3-x\right)\left(x+3\right)}-\frac{\left(3-x\right)^2}{\left(x+3\right)\left(3-x\right)}-\frac{4x^2}{\left(x+3\right)\left(3-x\right)}\right]\)\(:\frac{3x+4}{x+3}\).
\(P=\frac{\left(x+3\right)^2-\left(3-x\right)^2-4x^2}{\left(x+3\right)\left(3-x\right)}.\frac{x+3}{3x+4}\).
\(P=\frac{\left[\left(x+3\right)^2-\left(x-3\right)^2\right]-4x^2}{\left(x+3\right)\left(3-x\right)}.\frac{x+3}{3x+4}\).
\(P=\frac{\left[\left(x+3+x-3\right)\left(x+3-x+3\right)\right]-4x^2}{\left(x+3\right)\left(3-x\right)}.\frac{x+3}{3x+4}\).
\(P=\frac{2x.6-4x^2}{\left(x+3\right)\left(3-x\right)}.\frac{x+3}{3x+4}\).
\(P=\frac{\left(12x-4x^2\right)\left(x+3\right)}{\left(x+3\right)\left(3-x\right)\left(3x+4\right)}=\frac{4x\left(3-x\right)}{\left(3-x\right)\left(3x+4\right)}=\frac{4x}{3x+4}\).
Vậy với \(x\ne\pm3\)thì \(P=\frac{4x}{3x+4}\).
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