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a. \({{11x + 13} \over {3x – 3}} + {{15x + 17} \over {4 – 4x}}\)
b. \({{2x + 1} \over {2{x^2} – x}} + {{32{x^2}} \over {1 – 4{x^2}}} + {{1 – 2x} \over {2{x^2} + x}}\)
c. \({1 \over {{x^2} + x + 1}} + {1 \over {{x^2} – x}} + {{2x} \over {1 – {x^3}}}\)
d. \({{{x^4}} \over {1 – x}} + {x^3} + {x^2} + x + 1\)
a. \({{11x + 13} \over {3x – 3}} + {{15x + 17} \over {4 – 4x}}\)\( = {{11x + 13} \over {3\left( {x – 1} \right)}} + {{ – 15x – 17} \over {4\left( {x – 1} \right)}}\)
\(\eqalign{ & = {{4\left( {11x + 13} \right)} \over {12\left( {x – 1} \right)}} + {{3\left( { – 15x – 17} \right)} \over {12\left( {x – 1} \right)}} = {{44x + 52 – 45x – 51} \over {12\left( {x – 1} \right)}} = {{1 – x} \over {12\left( {x – 1} \right)}} \cr & = {{ – \left( {x – 1} \right)} \over {12\left( {x – 1} \right)}} = – {1 \over {12}} \cr} \)
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b. \({{2x + 1} \over {2{x^2} – x}} + {{32{x^2}} \over {1 – 4{x^2}}} + {{1 – 2x} \over {2{x^2} + x}}\)\( = {{2x + 1} \over {x\left( {2x – 1} \right)}} + {{ – 32{x^2}} \over {\left( {2x + 1} \right)\left( {2x – 1} \right)}} + {{1 – 2x} \over {x\left( {2x + 1} \right)}}\)
\(\eqalign{ & = {{\left( {2x + 1} \right)\left( {2x + 1} \right)} \over {x\left( {2x + 1} \right)\left( {2x – 1} \right)}} + {{ – 32{x^2}.x} \over {x\left( {2x + 1} \right)\left( {2x – 1} \right)}} + {{\left( {1 – 2x} \right)\left( {2x – 1} \right)} \over {x\left( {2x + 1} \right)\left( {2x – 1} \right)}} \cr & = {{4{x^2} + 4x + 1 – 32{x^3} + 2x – 1 – 4{x^2} + 2x} \over {x\left( {2x + 1} \right)\left( {2x – 1} \right)}} = {{ – 32{x^3} + 8x} \over {x\left( {2x + 1} \right)\left( {2x – 1} \right)}} \cr & = {{ – 8x\left( {4{x^2} – 1} \right)} \over {x\left( {2x + 1} \right)\left( {2x – 1} \right)}} = {{ – 8x\left( {2x + 1} \right)\left( {2x – 1} \right)} \over {x\left( {2x + 1} \right)\left( {2x – 1} \right)}} = – 8 \cr} \)
c. \({1 \over {{x^2} + x + 1}} + {1 \over {{x^2} – x}} + {{2x} \over {1 – {x^3}}}\)\( = {1 \over {{x^2} + x + 1}} + {1 \over {x\left( {x – 1} \right)}} + {{ – 2x} \over {\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}}\)
\(\eqalign{ & = {{x\left( {x – 1} \right)} \over {x\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}} + {{{x^2} + x + 1} \over {x\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}} + {{ – 2x.x} \over {x\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}} \cr & = {{{x^2} – x + {x^2} + x + 1 – 2{x^2}} \over {x\left( {x – 1} \right)\left( {{x^2} + x + 1} \right)}} = {1 \over {x\left( {{x^3} – 1} \right)}} \cr} \)
d. \({{{x^4}} \over {1 – x}} + {x^3} + {x^2} + x + 1\)\( = {{{x^4}} \over {1 – x}} + {{\left( {{x^3} + {x^2} + x + 1} \right)\left( {1 – x} \right)} \over {1 – x}}\)
\( = {{{x^4} + {x^3} + {x^2} + x + 1 – {x^4} – {x^3} – {x^2} – x} \over {1 – x}} = {1 \over {1 – x}}\)