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Section 5.5 : Partial Fractions

Determine the partial fraction decomposition of each of the following expressions.

1. \( \displaystyle \frac{{22 + 7x}}{{{x^2} + 5x + 4}}\)
2. \( \displaystyle \frac{{7x - 44}}{{4{x^2} + 25x - 21}}\)
3. \( \displaystyle \frac{{ - x - 47}}{{{x^2} - 11x + 24}}\)
4. \( \displaystyle \frac{{5 - 38x}}{{8{x^2} + 2x - 1}}\)
5. \( \displaystyle \frac{{6{x^2} + 50x + 16}}{{\left( {x - 1} \right)\left( {x + 2} \right)\left( {x + 7} \right)}}\)
6. \( \displaystyle \frac{{32{x^2} + 39x - 8}}{{\left( {x + 1} \right)\left( {x + 2} \right)\left( {2x - 3} \right)}}\)
7. \( \displaystyle \frac{{36 + 115x - 19{x^2}}}{{\left( {x + 3} \right)\left( {x - 5} \right)\left( {4x - 3} \right)}}\)
8. \( \displaystyle \frac{{3 - 5x}}{{{{\left( {x - 3} \right)}^2}}}\)
9. \( \displaystyle \frac{{24x + 41}}{{{{\left( {3x + 5} \right)}^2}}}\)
10. \( \displaystyle \frac{{10x + 93}}{{{{\left( {x + 10} \right)}^2}}}\)
11. \( \displaystyle \frac{{7{x^2} + 31x + 107}}{{\left( {x - 4} \right){{\left( {x + 3} \right)}^2}}}\)
12. \( \displaystyle \frac{{9{x^2} - 58x - 37}}{{\left( {x + 7} \right){{\left( {x - 2} \right)}^2}}}\)
13. \( \displaystyle \frac{{21{x^2} - 43x + 20}}{{\left( {3x - 2} \right){{\left( {x - 1} \right)}^2}}}\)
14. \( \displaystyle \frac{{ - 7{x^2} + 108x - 11}}{{x\left( {{x^2} - 9x + 1} \right)}}\)
15. \( \displaystyle \frac{{24{x^2} + 2x + 117}}{{x\left( {2{x^2} + x + 13} \right)}}\)
16. \( \displaystyle \frac{{2 - 11x + {x^2} - 7{x^3}}}{{{{\left( {{x^2} + 2} \right)}^2}}}\)
17. \( \displaystyle \frac{{4{x^3} - 3{x^2} - 5x - 5}}{{{{\left( {{x^2} + 1} \right)}^2}}}\)