Biến đổi các phân thức hữu tỉ

Ta có \(M = \left( {\dfrac{{x + 1}}{{x - 1}} - \dfrac{{x - 1}}{{x + 1}}} \right):\dfrac{{4x}}{{3x - 3}}\)   ĐK: \(x \ne  \pm 1\)

\( = \left[ {\dfrac{{{{\left( {x + 1} \right)}^2}}}{{\left( {x - 1} \right)\left( {x + 1} \right)}} - \dfrac{{{{\left( {x - 1} \right)}^2}}}{{\left( {x - 1} \right)\left( {x + 1} \right)}}} \right]:\dfrac{{4x}}{{3\left( {x - 1} \right)}}\) \( = \dfrac{{{x^2} + 2x + 1 - {x^2} + 2x - 1}}{{\left( {x - 1} \right)\left( {x + 1} \right)}}.\dfrac{{3\left( {x - 1} \right)}}{{4x}}\) \( = \dfrac{{4x}}{{\left( {x - 1} \right)\left( {x + 1} \right)}}.\dfrac{{3\left( {x - 1} \right)}}{{4x}} = \dfrac{3}{{x + 1}}\) .

Vậy \(M = \dfrac{3}{{x + 1}}\) với \(x \ne  \pm 1\) .

Ta có: Q = \(\dfrac{{{x^2} + 1}}{3}\)  với \(x \ne 0;x \ne  \pm 1\).

Ta có: \({x^2} \ge 4\,\,\forall x \ge 2 \Rightarrow {x^2} + 1 \ge 5\,\,\forall x \ge 2\)\( \Rightarrow \dfrac{{{x^2} + 1}}{3} \ge \dfrac{5}{3} \,\,\forall x \ge 2\).

Dấu “=” xảy ra khi \(x = 2\left( {tm} \right)\).

Vậy  \(Min\,\,Q = \dfrac{5}{3} \Leftrightarrow x = 2\).

Ta có \(M = \left( {\dfrac{{x + 1}}{{x - 1}} - \dfrac{{x - 1}}{{x + 1}}} \right):\dfrac{{4x}}{{3x - 3}}\)   ĐK: \(x \ne  \pm 1\)

\( = \left[ {\dfrac{{{{\left( {x + 1} \right)}^2}}}{{\left( {x - 1} \right)\left( {x + 1} \right)}} - \dfrac{{{{\left( {x - 1} \right)}^2}}}{{\left( {x - 1} \right)\left( {x + 1} \right)}}} \right]:\dfrac{{4x}}{{3\left( {x - 1} \right)}}\) \( = \dfrac{{{x^2} + 2x + 1 - {x^2} + 2x - 1}}{{\left( {x - 1} \right)\left( {x + 1} \right)}}.\dfrac{{3\left( {x - 1} \right)}}{{4x}}\) \( = \dfrac{{4x}}{{\left( {x - 1} \right)\left( {x + 1} \right)}}.\dfrac{{3\left( {x - 1} \right)}}{{4x}} = \dfrac{3}{{x + 1}}\) .

Vậy \(M = \dfrac{3}{{x + 1}}\) với \(x \ne  \pm 1\) .

ĐK: \(\left\{ \begin{array}{l}3x + {\left( {x - 1} \right)^2} \ne 0\\{x^3} - 1 \ne 0\\{x^3} + x \ne 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x \ne  \pm 1\\x \ne 0\end{array} \right..\)

\(Q = \left[ {\dfrac{{{{(x - 1)}^2}}}{{3x + {{(x - 1)}^2}}} - \dfrac{{1 - 2{x^2} + 4x}}{{{x^3} - 1}} + \dfrac{1}{{x - 1}}} \right]:\dfrac{{3x}}{{{x^3} + x}}\)

\(= \left[ {\dfrac{{{{\left( {x - 1} \right)}^2}}}{{3x + {x^2} - 2x + 1}} - \dfrac{{1 - 2{x^2} + 4x}}{{\left( {x - 1} \right)\left( {{x^2} + x + 1} \right)}} + \dfrac{1}{{x - 1}}} \right]:\dfrac{{3x}}{{x\left( {{x^2} + 1} \right)}}\)

\( = \left[ {\dfrac{{{{\left( {x - 1} \right)}^2}}}{{{x^2} + x + 1}} + \dfrac{{2{x^2} - 4x - 1}}{{\left( {x - 1} \right)\left( {{x^2} + x + 1} \right)}} + \dfrac{1}{{x - 1}}} \right].\dfrac{{x({x^2} + 1)}}{{3x}}\)

\( = \dfrac{{{{(x - 1)}^3} + 2{x^2} - 4x - 1 + {x^2} + x + 1}}{{{x^3} - 1}}.\dfrac{{{x^2} + 1}}{3}\)\( = \dfrac{{{x^3} - 3x{}^2 + 3x - 1 + 2{x^2} - 4x - 1 + {x^2} + x + 1}}{{{x^3} - 1}}.\dfrac{{{x^2} + 1}}{3}\)\( = \dfrac{{{x^3} - 1}}{{{x^3} - 1}}.\dfrac{{{x^2} + 1}}{3} = \dfrac{{{x^2} + 1}}{3}\)

Vậy \(Q = \dfrac{{{x^2} + 1}}{3}\)  với \(x \ne  \pm 1;x \ne 0\).

ĐK: \(\left\{ \begin{array}{l}3x + {\left( {x - 1} \right)^2} \ne 0\\{x^3} - 1 \ne 0\\{x^3} + x \ne 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x \ne  \pm 1\\x \ne 0\end{array} \right..\)

\(Q = \left[ {\dfrac{{{{(x - 1)}^2}}}{{3x + {{(x - 1)}^2}}} - \dfrac{{1 - 2{x^2} + 4x}}{{{x^3} - 1}} + \dfrac{1}{{x - 1}}} \right]:\dfrac{{3x}}{{{x^3} + x}}\)

\(= \left[ {\dfrac{{{{\left( {x - 1} \right)}^2}}}{{3x + {x^2} - 2x + 1}} - \dfrac{{1 - 2{x^2} + 4x}}{{\left( {x - 1} \right)\left( {{x^2} + x + 1} \right)}} + \dfrac{1}{{x - 1}}} \right]:\dfrac{{3x}}{{x\left( {{x^2} + 1} \right)}}\)

\( = \left[ {\dfrac{{{{\left( {x - 1} \right)}^2}}}{{{x^2} + x + 1}} + \dfrac{{2{x^2} - 4x - 1}}{{\left( {x - 1} \right)\left( {{x^2} + x + 1} \right)}} + \dfrac{1}{{x - 1}}} \right].\dfrac{{x({x^2} + 1)}}{{3x}}\)

\( = \dfrac{{{{(x - 1)}^3} + 2{x^2} - 4x - 1 + {x^2} + x + 1}}{{{x^3} - 1}}.\dfrac{{{x^2} + 1}}{3}\)\( = \dfrac{{{x^3} - 3x{}^2 + 3x - 1 + 2{x^2} - 4x - 1 + {x^2} + x + 1}}{{{x^3} - 1}}.\dfrac{{{x^2} + 1}}{3}\)\( = \dfrac{{{x^3} - 1}}{{{x^3} - 1}}.\dfrac{{{x^2} + 1}}{3} = \dfrac{{{x^2} + 1}}{3}\)

Vậy \(Q = \dfrac{{{x^2} + 1}}{3}\)  với \(x \ne  \pm 1;x \ne 0\).

Phân thức \(\dfrac{{{x^2} - 4x + 4}}{{x - 2}}\) xác định khi \(x - 2 \ne 0 \Leftrightarrow x \ne 2\) .

Phân thức \(B = \left( {\dfrac{1}{{x - 2}} - \dfrac{{2x}}{{4 - {x^2}}} + \dfrac{1}{{2 + x}}} \right):\left( {\dfrac{2}{x} - 1} \right)\) xác định khi

\(\left\{ \begin{array}{l}x - 2 \ne 0\\4 - {x^2} \ne 0\\2 + x \ne 0\\x \ne 0\end{array} \right.\)\( \Leftrightarrow \left\{ \begin{array}{l}x \ne 2\\x \ne  - 2\\x \ne 0\\{x^2} \ne 4\end{array} \right.\) \( \Leftrightarrow \left\{ \begin{array}{l}x \ne 2\\x \ne  - 2\\x \ne 0\end{array} \right.\)  .

Ta có  \(C = \left( {\dfrac{{21}}{{{x^2} - 9}} - \dfrac{{x - 4}}{{3 - x}} - \dfrac{{x - 1}}{{3 + x}}} \right):\left( {1 - \dfrac{1}{{x + 3}}} \right)\)

\( = \left[ {\dfrac{{21}}{{\left( {x - 3} \right)\left( {x + 3} \right)}} + \dfrac{{\left( {x - 4} \right)\left( {x + 3} \right)}}{{\left( {x - 3} \right)\left( {x + 3} \right)}} - \dfrac{{\left( {x - 1} \right)\left( {x - 3} \right)}}{{\left( {x - 3} \right)\left( {x + 3} \right)}}} \right]:\left( {\dfrac{{x + 3 - 1}}{{x + 3}}} \right)\)   Điều kiện: \(x \ne  \pm 3\)

\( = \dfrac{{21 + {x^2} - x - 12 - {x^2} + 4x - 3}}{{\left( {x - 3} \right)\left( {x + 3} \right)}}:\dfrac{{x + 2}}{{x + 3}}\)

\( = \dfrac{{3x + 6}}{{\left( {x - 3} \right)\left( {x + 3} \right)}}.\dfrac{{x + 3}}{{x + 2}} = \dfrac{{3\left( {x + 2} \right)}}{{\left( {x - 3} \right)\left( {x + 3} \right)}}.\dfrac{{x + 3}}{{x + 2}}\)

\( = \dfrac{3}{{x - 3}}\). Vậy \(C = \dfrac{3}{{x - 3}}\) .

Ta có  \(P = \left( {\dfrac{x}{{x + 2}} - \dfrac{{{x^3} - 8}}{{{x^3} + 8}}.\dfrac{{{x^2} - 2x + 4}}{{{x^2} - 4}}} \right):\dfrac{4}{{x + 2}}\)

\( = \left( {\dfrac{x}{{x + 2}} - \dfrac{{\left( {x - 2} \right)\left( {{x^2} + 2x + 4} \right)}}{{\left( {x + 2} \right)\left( {{x^2} - 2x + 4} \right)}}.\dfrac{{{x^2} - 2x + 4}}{{\left( {x - 2} \right)\left( {x + 2} \right)}}} \right).\dfrac{{x + 2}}{4}\)      ĐK: \(x \ne  \pm 2\)

\( = \left( {\dfrac{x}{{x + 2}} - \dfrac{{{x^2} + 2x + 4}}{{{{\left( {x + 2} \right)}^2}}}} \right).\dfrac{{x + 2}}{4}\) \( = \left[ {\dfrac{{x\left( {x + 2} \right) - {x^2} - 2x - 4}}{{{{\left( {x + 2} \right)}^2}}}} \right].\dfrac{{x + 2}}{4}\)

\( = \dfrac{{ - 4}}{{{{\left( {x + 2} \right)}^2}}}.\dfrac{{x + 2}}{4}\) \( = \dfrac{{ - 1}}{{x + 2}}\) .

Vậy $P = \dfrac{{ - 1}}{{x + 2}}$ .

Ta có \(M = \left( {\dfrac{{x + 1}}{{x - 1}} - \dfrac{{x - 1}}{{x + 1}}} \right):\dfrac{{4x}}{{3x - 3}}\)   ĐK: \(x \ne  \pm 1\)

\( = \left[ {\dfrac{{{{\left( {x + 1} \right)}^2}}}{{\left( {x - 1} \right)\left( {x + 1} \right)}} - \dfrac{{{{\left( {x - 1} \right)}^2}}}{{\left( {x - 1} \right)\left( {x + 1} \right)}}} \right]:\dfrac{{4x}}{{3\left( {x - 1} \right)}}\) \( = \dfrac{{{x^2} + 2x + 1 - {x^2} + 2x - 1}}{{\left( {x - 1} \right)\left( {x + 1} \right)}}.\dfrac{{3\left( {x - 1} \right)}}{{4x}}\) \( = \dfrac{{4x}}{{\left( {x - 1} \right)\left( {x + 1} \right)}}.\dfrac{{3\left( {x - 1} \right)}}{{4x}} = \dfrac{3}{{x + 1}}\) .

Vậy \(M = \dfrac{3}{{x + 1}}\) với \(x \ne  \pm 1\) .

Ta có \(\dfrac{{1 + \dfrac{1}{x}}}{{x - \dfrac{1}{x}}}\)\( = \dfrac{{\dfrac{{x + 1}}{x}}}{{\dfrac{{{x^2} - 1}}{x}}}\) \( = \dfrac{{x + 1}}{x}:\dfrac{{{x^2} - 1}}{x} = \dfrac{{x + 1}}{x}.\dfrac{x}{{{x^2} - 1}} = \dfrac{{x + 1}}{{\left( {x + 1} \right)\left( {x - 1} \right)}} = \dfrac{1}{{x - 1}}\) .

Ta có \(\dfrac{{x + \dfrac{1}{{{x^2}}}}}{{1 - \dfrac{1}{x} + \dfrac{1}{{{x^2}}}}}\)\( = \dfrac{{\dfrac{{{x^3}}}{{{x^2}}} + \dfrac{1}{{{x^2}}}}}{{\dfrac{{{x^2}}}{{{x^2}}} - \dfrac{x}{{{x^2}}} + \dfrac{1}{{{x^2}}}}} = \dfrac{{\dfrac{{{x^3} + 1}}{{{x^2}}}}}{{\dfrac{{{x^2} - x + 1}}{{{x^2}}}}}\) \( = \dfrac{{{x^3} + 1}}{{{x^2}}}:\dfrac{{{x^2} - x + 1}}{{{x^2}}} = \dfrac{{{x^3} + 1}}{{{x^2}}}.\dfrac{{{x^2}}}{{{x^2} - x + 1}}\)

\( = \dfrac{{\left( {x + 1} \right)\left( {{x^2} - x + 1} \right)}}{{{x^2} - x + 1}} = x + 1\) .

Ta có \(\left( {\dfrac{1}{{{x^2} + 4x + 4}} - \dfrac{1}{{{x^2} - 4x + 4}}} \right):\left( {\dfrac{1}{{x + 2}} + \dfrac{1}{{x - 2}}} \right)\)\( = \left( {\dfrac{1}{{{{\left( {x + 2} \right)}^2}}} - \dfrac{1}{{{{\left( {x - 2} \right)}^2}}}} \right):\left( {\dfrac{{x - 2 + x + 2}}{{\left( {x - 2} \right)\left( {x + 2} \right)}}} \right)\)

$ = \dfrac{{{x^2} - 4x + 4 - \left( {{x^2} + 4x + 4} \right)}}{{{{\left( {x - 2} \right)}^2}{{\left( {x + 2} \right)}^2}}}.\dfrac{{\left( {x + 2} \right)\left( {x - 2} \right)}}{{2x}}$\( = \dfrac{{ - 8x}}{{{{\left( {x - 2} \right)}^2}{{\left( {x + 2} \right)}^2}}}.\dfrac{{\left( {x - 2} \right)\left( {x + 2} \right)}}{{2x}} = \dfrac{{ - 4}}{{\left( {x + 2} \right)\left( {x - 2} \right)}} = \dfrac{{ - 4}}{{{x^2} - 4}}\)

Ta có \(A = \left( {\dfrac{1}{{{x^2} + x}} - \dfrac{{2 - x}}{{x + 1}}} \right):\left( {\dfrac{1}{x} + x - 2} \right)\)\( = \left( {\dfrac{1}{{x\left( {x + 1} \right)}} - \dfrac{{x\left( {2 - x} \right)}}{{x\left( {x + 1} \right)}}} \right):\left( {\dfrac{1}{x} + \dfrac{{{x^2}}}{x} - \dfrac{{2x}}{x}} \right)\) \( = \dfrac{{1 - 2x + {x^2}}}{{x\left( {x + 1} \right)}}:\dfrac{{1 + {x^2} - 2x}}{x}\)

\( = \dfrac{{{x^2} - 2x + 1}}{{x\left( {x + 1} \right)}}.\dfrac{x}{{{x^2} - 2x + 1}} = \dfrac{1}{{x + 1}}\) .

Vậy số cần điền là \(1\) .

Phân thức \(\dfrac{{{x^2} - 4x + 4}}{{x - 2}}\) xác định khi \(x - 2 \ne 0 \Leftrightarrow x \ne 2\) .

Ta có \(\dfrac{{{x^2} - 4x + 4}}{{x - 2}}\)\( = \dfrac{{{{\left( {x - 2} \right)}^2}}}{{x - 2}} = x - 2\)

Thay \(x = 2020\) (thỏa mãn điều kiện \(x \ne 2\) ) vào biểu thức \(x - 2\) ta được \(2020 - 2 = 2018\) .

Vậy với  \(x = 2020\) thì giá trị biểu thức là \(2018\) .

Phân thức \(B = \left( {\dfrac{1}{{x - 2}} - \dfrac{{2x}}{{4 - {x^2}}} + \dfrac{1}{{2 + x}}} \right):\left( {\dfrac{2}{x} - 1} \right)\) xác định khi

\(\left\{ \begin{array}{l}x - 2 \ne 0\\4 - {x^2} \ne 0\\2 + x \ne 0\\x \ne 0\end{array} \right.\)\( \Leftrightarrow \left\{ \begin{array}{l}x \ne 2\\x \ne  - 2\\x \ne 0\\{x^2} \ne 4\end{array} \right.\) \( \Leftrightarrow \left\{ \begin{array}{l}x \ne 2\\x \ne  - 2\\x \ne 0\end{array} \right.\)  .

Phân thức \(B = \left( {\dfrac{1}{{x - 2}} - \dfrac{{2x}}{{4 - {x^2}}} + \dfrac{1}{{2 + x}}} \right).\left( {\dfrac{2}{x} - 1} \right)\)

\( = \left( {\dfrac{1}{{x - 2}} + \dfrac{{2x}}{{\left( {x - 2} \right)\left( {x + 2} \right)}} + \dfrac{1}{{x + 2}}} \right).\left( {\dfrac{{2 - x}}{x}} \right)\)

$ = \left( {\dfrac{{x + 2}}{{\left( {x - 2} \right)\left( {x + 2} \right)}} + \dfrac{{2x}}{{\left( {x - 2} \right)\left( {x + 2} \right)}} + \dfrac{{x - 2}}{{\left( {x - 2} \right)\left( {x + 2} \right)}}} \right).\left[ {\dfrac{{ - \left( {x - 2} \right)}}{x}} \right]$

\( = \dfrac{{x + 2 + 2x + x - 2}}{{\left( {x - 2} \right)\left( {x + 2} \right)}}.\dfrac{{\left[ { - \left( {x - 2} \right)} \right]}}{x}\)

$ = \dfrac{{4x}}{{\left( {x - 2} \right)\left( {x + 2} \right)}}.\dfrac{{\left[ { - \left( {x - 2} \right)} \right]}}{x} = \dfrac{{ - 4}}{{x + 2}}$

Vậy \(B = \dfrac{{ - 4}}{{x + 2}}\) .

Theo câu trước ta có  \(B = \dfrac{{ - 4}}{{x + 2}}\) với \(x \ne \left\{ { - 2;0;2} \right\}\).

Ta có \(B = \dfrac{1}{2}\) \( \Leftrightarrow \) \(\dfrac{{ - 4}}{{x + 2}} = \dfrac{1}{2} \Leftrightarrow \dfrac{{ - 8}}{{2\left( {x + 2} \right)}} = \dfrac{{x + 2}}{{2\left( {x + 2} \right)}} \Rightarrow x + 2 =  - 8 \Leftrightarrow x =  - 10\,\left( {TM} \right)\).

Vậy \(x =  - 10\) .

Theo các câu trước ta có  \(B = \dfrac{{ - 4}}{{x + 2}}\) với \(x \ne \left\{ { - 2;0;2} \right\}\).

Để \(B > 0 \Leftrightarrow \dfrac{{ - 4}}{{x + 2}} > 0\) mà \( - 4 < 0 \Rightarrow x + 2 < 0 \Leftrightarrow x <  - 2\) .

Kết hợp điều kiện \(x \ne \left\{ { - 2;0;2} \right\}\) ta có \(x <  - 2\) .