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Khẳng định `A` là đúng vì :
\(\dfrac{\left(x-1\right)^2}{x-2}\\ =\dfrac{\left(x-1\right)^2}{-\left(x-2\right)}\\ =\dfrac{\left(1-x\right)^2}{2-x}\)
`->` Đã là hằng đẳng thức mũ `2` thì `(x-1)^2=(1-x)^2`
a)
\(\begin{array}{l}\frac{2}{{3{\rm{x}}}} + \frac{x}{{x - 1}} + \frac{{6{{\rm{x}}^2} - 4}}{{2{\rm{x}}\left( {1 - x} \right)}}\\ = \frac{2}{{3{\rm{x}}}} - \frac{x}{{1 - x}} + \frac{{6{{\rm{x}}^2} - 4}}{{2{\rm{x}}\left( {1 - x} \right)}}\\ = \frac{{4\left( {1 - x} \right) - 6{{\rm{x}}^2} + 3\left( {6{{\rm{x}}^2} - 4} \right)}}{{6{\rm{x}}\left( {1 - x} \right)}}\\ = \frac{{4 - 4{\rm{x}} - 6{{\rm{x}}^2} + 18{{\rm{x}}^2} - 12}}{{6{\rm{x}}\left( {1 - x} \right)}}\\ = \frac{{12{{\rm{x}}^2} - 4{\rm{x}} - 8}}{{6{\rm{x}}\left( {1 - x} \right)}}\end{array}\)
b)
\(\begin{array}{l}\frac{{{x^3} + 1}}{{1 - {x^3}}} + \frac{x}{{x - 1}} - \frac{{x + 1}}{{{x^2} + x + 1}}\\ = \frac{{ - {x^3} - 1}}{{{x^3} - 1}} + \frac{x}{{x - 1}} - \frac{{x + 1}}{{{x^2} + x + 1}}\\ = \frac{{ - {x^3} - 1 + x\left( {{x^2} + x + 1} \right) - \left( {{x^2} - 1} \right)}}{{\left( {x - 1} \right)\left( {{x^2} + x + 1} \right)}}\\ = \frac{{ - {x^3} - 1 + {x^3} + {x^2} + x - {x^2} + 1}}{{\left( {x - 1} \right)\left( {{x^2} + x + 1} \right)}}\\ = \frac{x}{{{x^3} - 1}}\end{array}\)
c)
\(\begin{array}{l}\left( {\frac{2}{{x + 2}} - \frac{2}{{1 - x}}} \right).\frac{{{x^2} - 4}}{{4{{\rm{x}}^2} - 1}}\\ = \frac{{2\left( {1 - x} \right) - 2\left( {x + 2} \right)}}{{\left( {x + 2} \right)\left( {1 - x} \right)}}.\frac{{{x^2} - 4}}{{4{{\rm{x}}^2} - 1}}\\ = \frac{{2 - 2{\rm{x}} - 2{\rm{x}} - 4}}{{\left( {x + 2} \right)\left( {1 - x} \right)}}.\frac{{{x^2} - 4}}{{4{{\rm{x}}^2} - 1}}\\ = \frac{{ - 4{\rm{x - 2}}}}{{\left( {x + 2} \right)\left( {1 - x} \right)}}.\frac{{{x^2} - 4}}{{4{{\rm{x}}^2} - 1}}\\ = \frac{{\left( { - 4{\rm{x}} - 2} \right)\left( {x - 2} \right)\left( {x + 2} \right)}}{{\left( {x + 2} \right)\left( {1 - x} \right)\left( {2{\rm{x}} - 1} \right)\left( {2{\rm{x}} + 1} \right)}}\\ = \frac{{ - 4{{\rm{x}}^2} + 8{\rm{x}} - 2{\rm{x}} + 4}}{{\left( {1 - x} \right)\left( {2{\rm{x}} - 1} \right)\left( {2{\rm{x}} + 1} \right)}}\\ = \frac{{ - 4{{\rm{x}}^2} + 6{\rm{x}} + 4}}{{\left( {1 - x} \right)\left( {4{{\rm{x}}^2} - 1} \right)}}\end{array}\)
d)
\(\begin{array}{l}1 + \frac{{{x^3} - x}}{{{x^2} + 1}}\left( {\frac{1}{{1 - x}} - \frac{1}{{1 - {x^2}}}} \right)\\ = 1 + \frac{{{x^3} - x}}{{{x^2} + 1}}\left( {\frac{1}{{1 - x}} - \frac{1}{{1 - {x^2}}}} \right)\\ = 1 + \frac{{{x^3} - x}}{{{x^2} + 1}}.\frac{{1 + x - 1}}{{1 - {x^2}}}\\ = 1 + \frac{{x\left( {{x^2} - 1} \right)}}{{{x^2} + 1}}.\frac{x}{{1 - {x^2}}}\\ = 1 + \frac{{ - {x^2}\left( {{x^2} - 1} \right)}}{{\left( {{x^2} + 1} \right)\left( {{x^2} - 1} \right)}}\\ = 1 + \frac{{ - {x^2}}}{{{x^2} + 1}}\\ = \frac{{{x^2} + 1 - {x^2}}}{{{x^2} + 1}}\\ = \frac{1}{{{x^2} + 1}}\end{array}\)
\(\begin{array}{l}a)\frac{{4{\rm{x}} - 6}}{{5{{\rm{x}}^2} - x}}.\frac{{25{{\rm{x}}^2} - 10{\rm{x}} + 1}}{{27 + 8{{\rm{x}}^3}}}\\ = \frac{{ - 2\left( {3 - 2{\rm{x}}} \right)}}{{x\left( {5{\rm{x}} - 1} \right)}}.\frac{{{{\left( {5{\rm{x}} - 1} \right)}^2}}}{{\left( {3 - 2{\rm{x}}} \right)\left( {9 + 6{\rm{x}} + 4{{\rm{x}}^2}} \right)}}\\ = \frac{{ - 2\left( {5{\rm{x}} - 1} \right)}}{{x\left( {9 + 6{\rm{x}} + 4{{\rm{x}}^2}} \right)}}\\b)\frac{{2{\rm{x}} + 10}}{{{{\left( {x - 3} \right)}^2}}}:\frac{{{{\left( {x + 5} \right)}^3}}}{{{x^2} - 9}}\\ = \frac{{2{\rm{x}} + 10}}{{{{\left( {x - 3} \right)}^2}}}.\frac{{{x^2} - 9}}{{{{\left( {x + 5} \right)}^2}}}\\ = \frac{{2\left( {x + 5} \right)\left( {x - 3} \right)\left( {x + 3} \right)}}{{{{\left( {x - 3} \right)}^2}{{\left( {x + 5} \right)}^3}}}\\ = \frac{{2\left( {x + 3} \right)}}{{\left( {x - 3} \right){{\left( {x + 5} \right)}^2}}}\end{array}\)
\(a)\left( { - \frac{{3{\rm{x}}}}{{5{\rm{x}}{y^2}}}} \right):\left( { - \frac{{5{y^2}}}{{12{\rm{x}}y}}} \right) = \frac{{ - 3{\rm{x}}}}{{5{\rm{x}}{y^2}}}.\frac{{ - 12{\rm{x}}y}}{{5{y^2}}} = \frac{{36{{\rm{x}}^2}y}}{{25{\rm{x}}{y^4}}}\)
b) \(\frac{4{{\text{x}}^{2}}-1}{8{{\text{x}}^{3}}-1}:\frac{4{{\text{x}}^{2}}+4\text{x}+1}{4{{\text{x}}^{2}}+2\text{x}+1}=\frac{4{{\text{x}}^{2}}-1}{8{{\text{x}}^{3}}-1}.\frac{4{{\text{x}}^{2}}+2\text{x}+1}{4{{\text{x}}^{2}}+4\text{x}+1}\)
\(=\frac{\left( 2\text{x}-1 \right)\left( 2\text{x}+1 \right)\left( 4{{\text{x}}^{2}}+2\text{x}+1 \right)}{\left( 2\text{x}-1 \right)\left( 4{{\text{x}}^{2}}+2\text{x}+1 \right){{\left( 2\text{x}+1 \right)}^{2}}}=\frac{1}{2\text{x}+1}\).
\(\begin{array}{l}a)\frac{{{x^2} + 4{\rm{x}} + 4}}{{{x^2} - 4}} + \frac{x}{{2 - x}} + \frac{{4 - x}}{{5{\rm{x}} - 10}}\\ = \frac{{{{\left( {x + 2} \right)}^2}}}{{\left( {x + 2} \right)\left( {x - 2} \right)}} - \frac{x}{{x - 2}} + \frac{{4 - x}}{{5\left( {x - 2} \right)}}\\ = \frac{{x + 2}}{{x - 2}} - \frac{x}{{x - 2}} + \frac{{4 - x}}{{5\left( {x - 2} \right)}}\\ = \frac{{5\left( {x + 2} \right) - 5x + 4 - x}}{{5\left( {x - 2} \right)}} = \frac{{ - x + 14}}{{5\left( {x - 2} \right)}}\end{array}\)
\(\begin{array}{l}b)\frac{x}{{{x^2} + 1}} - \left( {\frac{3}{{x + 6}} + \frac{{x - 2}}{{x + 4}}} \right) + \left[ {\frac{3}{{x + 6}} - \left( {\frac{1}{{{x^2} + 1}} - \frac{{x - 2}}{{x + 4}}} \right)} \right]\\ = \frac{x}{{{x^2} + 1}} - \frac{3}{{x + 6}} - \frac{{x - 2}}{{x + 4}} + \frac{3}{{x + 6}} - \frac{1}{{{x^2} + 1}} + \frac{{x - 2}}{{x + 4}}\\ = \frac{x}{{{x^2} + 1}} - \frac{1}{{{x^2} + 1}} = \frac{{x - 1}}{{{x^2} + 1}}\end{array}\)
\(a)\frac{{5{\rm{x}} + 10}}{{25{{\rm{x}}^2} + 50}} = \frac{{5\left( {x + 2} \right)}}{{25\left( {{x^2} + 2} \right)}} = \frac{{x + 2}}{{5\left( {{x^2} + 2} \right)}}\)
\(b)\frac{{45{\rm{x}}\left( {3 - x} \right)}}{{15{\rm{x}}{{\left( {x - 3} \right)}^2}}} = \frac{{3\left( {3 - x} \right)}}{{{{\left( {x - 3} \right)}^2}}}\)
\(c)\frac{{{{\left( {{x^2} - 1} \right)}^2}}}{{\left( {x + 1} \right)\left( {{x^3} + 1} \right)}} = \frac{{\left( {{x^2} - 1} \right)\left( {{x^2} - 1} \right)}}{{\left( {x + 1} \right)\left( {x + 1} \right)\left( {{x^2} - x + 1} \right)}} = \frac{{\left( {x + 1} \right)\left( {x - 1} \right)\left( {x + 1} \right)\left( {x - 1} \right)}}{{\left( {x + 1} \right)\left( {x + 1} \right)\left( {{x^2} - x + 1} \right)}} = \frac{{{{\left( {x - 1} \right)}^2}}}{{{x^2} - x + 1}}\)
a) Ta có: \(P = \frac{{x + 1}}{{{x^2} - 1}} = \frac{{x + 1}}{{\left( {x - 1} \right)\left( {x + 1} \right)}} = \frac{1}{{x - 1}}\)
Suy ra: \(Q = \frac{1}{{x - 1}}\)
b) Thay x = 11 vào P ta được: \(P = \frac{{11 + 1}}{{{{11}^2} - 1}} = \frac{1}{{10}}\)
Thay x = 11 vào Q ta được: \(Q = \frac{1}{{11 - 1}} = \frac{1}{{10}}\)
Hai kết quả P = Q tại x = 11
\(a)\left( { - \frac{{3{\rm{x}}}}{{5{\rm{x}}{y^2}}}} \right).\left( { - \frac{{5{y^2}}}{{12{\rm{x}}y}}} \right) = \frac{{\left( { - 3{\rm{x}}} \right).\left( { - 5{y^2}} \right)}}{{5{\rm{x}}{y^2}.12{\rm{x}}y}} = \frac{1}{{4{\rm{x}}y}}\)
\(b)\frac{{{x^2} - x}}{{2{\rm{x}} + 1}}.\frac{{4{{\rm{x}}^2} - 1}}{{{x^3} - 1}} = \frac{{x\left( {x - 1} \right).\left( {2{\rm{x}} - 1} \right)\left( {2{\rm{x}} + 1} \right)}}{{\left( {2{\rm{x}} + 1} \right).\left( {x - 1} \right)\left( {{x^2} + x + 1} \right)}} = \frac{{x\left( {2{\rm{x}} - 1} \right)}}{{{x^2} + x + 1}}\)
Khẳng định C là khẳng định sai vì:
Nếu: \(\frac{{x + 1}}{{x - 1}} = \frac{{{x^2} + x + 1}}{{{x^2} - x + 1}}\)
\(\begin{array}{l} \Rightarrow \frac{{x + 1}}{{x - 1}} - \frac{{{x^2} + x + 1}}{{{x^2} - x + 1}} = 0\\ \Rightarrow \frac{{\left( {x + 1} \right)\left( {{x^2} - x + 1} \right) - \left( {{x^2} + x + 1} \right)\left( {x - 1} \right)}}{{\left( {x - 1} \right)\left( {{x^2} - x + 1} \right)}} = 0\\ \Rightarrow \frac{{\left( {{x^3} + 1} \right) - \left( {{x^3} - 1} \right)}}{{\left( {x - 1} \right)\left( {{x^2} - x + 1} \right)}} = \frac{2}{{\left( {x - 1} \right)\left( {{x^2} - x + 1} \right)}} = 0\end{array}\)
\( \Rightarrow \) vô lý