Giải bài 6.42 trang 26 SGK Toán 8 tập 2 – Kết nối tri thức

Đề bài

Rút gọn biểu thức sau:

a) \(\frac{2}{{3{\rm{x}}}} + \frac{x}{{x – 1}} + \frac{{6{{\rm{x}}^2} – 4}}{{2{\rm{x}}\left( {1 – x} \right)}}\)

b) \(\frac{{{x^3} + 1}}{{1 – {x^3}}} + \frac{x}{{x – 1}} – \frac{{x + 1}}{{{x^2} + x + 1}}\)

c) \(\left( {\frac{2}{{x + 2}} – \frac{2}{{1 – x}}} \right).\frac{{{x^2} – 4}}{{4{{\rm{x}}^2} – 1}}\)

d) \(1 + \frac{{{x^3} – x}}{{{x^2} + 1}}\left( {\frac{1}{{1 – x}} – \frac{1}{{1 – {x^2}}}} \right)\)

Lời giải chi tiết

a) \(\frac{2}{{3{\rm{x}}}} + \frac{x}{{x – 1}} + \frac{{6{x^2} – 4}}{{2x\left( {1 – x} \right)}}\) \( = \frac{2}{{3{\rm{x}}}} + \frac{{ – x}}{{1 – x}} + \frac{{3{{\rm{x}}^2} – 2}}{{x\left( {1 – x} \right)}}\) \( = \frac{{2 – 2x – 3{x^2} + 9{x^2} – 6}}{{3x\left( {1 – x} \right)}}\) \( = \frac{{6{x^2} – 2x – 4}}{{3x\left( {1 – x} \right)}} \) \( = \frac{2({3x+1})}{3x} \)

b) \(\frac{{{x^3} + 1}}{{1 – {x^3}}} + \frac{x}{{x – 1}} – \frac{{x + 1}}{{{x^2} + x + 1}} \) \( = \frac{{ – {x^3} – 1}}{{{x^3} – 1}} + \frac{x}{{x – 1}} – \frac{{x + 1}}{{{x^2} + x + 1}} \) \( = \frac{{ – {x^3} – 1 + x\left( {{x^2} + x + 1} \right) – \left( {{x^2} – 1} \right)}}{{\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}} \) \( = \frac{{ – {x^3} – 1 + {x^3} + {x^2} + x – {x^2} + 1}}{{\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}} \) \( = \frac{x}{{{x^3} – 1}}\)

c) Ta có: 

\(\frac{2}{{x + 2}} – \frac{2}{{1 – x}} \) \( = \frac{{2\left( {1 – x} \right) – 2\left( {x + 2} \right)}}{{\left( {x + 2} \right)\left( {1 – x} \right)}} \) \( = \frac{{2 – 2{\rm{x}} – 2{\rm{x}} – 4}}{{\left( {x + 2} \right)\left( {1 – x} \right)}} \) \( = \frac{{ – 4x – 2}}{{\left( {x + 2} \right)\left( {1 – x} \right)}} \) \( = \frac{{2\left( {2x + 1} \right)}}{{\left( {x + 2} \right)\left( {x – 1} \right)}}\);

\(\frac{{{x^2} – 4}}{{4{{\rm{x}}^2} – 1}} \) \( = \frac{{\left( {x – 2} \right)\left( {x + 2} \right)}}{{\left( {2x – 1} \right)\left( {2x + 1} \right)}}\).

Do đó  \(\left( {\frac{2}{{x + 2}} – \frac{2}{{1 – x}}} \right).\frac{{{x^2} – 4}}{{4{{\rm{x}}^2} – 1}} \) \( = \frac{{2\left( {2x + 1} \right)}}{{\left( {x + 2} \right)\left( {x – 1} \right)}}.\frac{{\left( {x – 2} \right)\left( {x + 2} \right)}}{{\left( {2x – 1} \right)\left( {2x + 1} \right)}} \) \( = \frac{{2(x – 2)}}{{\left( {2x – 1} \right)\left( {x – 1} \right)}}\)

d) \(1 + \frac{{{x^3} – x}}{{{x^2} + 1}}\left( {\frac{1}{{1 – x}} – \frac{1}{{1 – {x^2}}}} \right) \) \( = 1 + \frac{{{x^3} – x}}{{{x^2} + 1}}\left( {\frac{1}{{1 – x}} – \frac{1}{{1 – {x^2}}}} \right) \) \( = 1 + \frac{{{x^3} – x}}{{{x^2} + 1}}.\frac{{1 + x – 1}}{{1 – {x^2}}} \) \( = 1 + \frac{{x\left( {{x^2} – 1} \right)}}{{{x^2} + 1}}.\frac{x}{{1 – {x^2}}} \) \( = 1 + \frac{{ – {x^2}\left( {{x^2} – 1} \right)}}{{\left( {{x^2} + 1} \right)\left( {{x^2} – 1} \right)}} \) \( = 1 + \frac{{ – {x^2}}}{{{x^2} + 1}} \) \( = \frac{{{x^2} + 1 – {x^2}}}{{{x^2} + 1}} \) \( = \frac{1}{{{x^2} + 1}}\)