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a. \({{1 – 2x} \over {6{x^3}y}} + {{3 + 2y} \over {6{x^3}y}} + {{2x – 4} \over {6{x^3}y}}\)
b. \({{{x^2} – 2} \over {x{{\left( {x – 1} \right)}^2}}} + {{2 – x} \over {x{{\left( {x – 1} \right)}^2}}}\)
c. \({{3x + 1} \over {{x^2} – 3x + 1}} + {{{x^2} – 6x} \over {{x^2} – 3x + 1}}\)
d. \({{{x^2} + 38x + 4} \over {2{x^2} + 17x + 1}} + {{3{x^2} – 4x – 2} \over {2{x^2} + 17x + 1}}\)
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a. \({{1 – 2x} \over {6{x^3}y}} + {{3 + 2y} \over {6{x^3}y}} + {{2x – 4} \over {6{x^3}y}}\) \( = {{1 – 2x + 3 + 2y + 2x – 4} \over {6{x^3}y}} = {{2y} \over {6{x^3}y}} = {1 \over {3{x^3}}}\)
b. \({{{x^2} – 2} \over {x{{\left( {x – 1} \right)}^2}}} + {{2 – x} \over {x{{\left( {x – 1} \right)}^2}}}\) \( = {{{x^2} – 2 + 2 – x} \over {x{{\left( {x – 1} \right)}^2}}} = {{x\left( {x – 1} \right)} \over {x{{\left( {x – 1} \right)}^2}}} = {1 \over {x – 1}}\)
c. \({{3x + 1} \over {{x^2} – 3x + 1}} + {{{x^2} – 6x} \over {{x^2} – 3x + 1}}\) \( = {{3x + 1 + {x^2} – 6x} \over {{x^2} – 3x + 1}} = {{{x^2} – 3x + 1} \over {{x^2} – 3x + 1}} = 1\)
d. \({{{x^2} + 38x + 4} \over {2{x^2} + 17x + 1}} + {{3{x^2} – 4x – 2} \over {2{x^2} + 17x + 1}}\) \( = {{{x^2} + 38x + 4 + 3{x^2} – 4x – 2} \over {2{x^2} + 17x + 1}} = {{4{x^2} + 34x + 2} \over {2{x^2} + 17x + 1}} = {{2\left( {2{x^2} + 17x + 1} \right)} \over {2{x^2} + 17x + 1}} = 2\)