Exercises 10 (Partial fractions)

Express the following fractions in terms of partial fractions:

1. \(\dfrac{x-4}{(x-2)(x+1)}\)
2. \(\dfrac{3x+2}{(x+1)(x-5)}\)
3. \(\dfrac{4x^2-2}{(x^2+3)(x-2)}\)
4. \(\dfrac{3x^2+1}{(x^2+4)(3x-1)}\)
5. \(\dfrac{4x^2-3x+2}{(x-2)(2+x^2)}\)
6. \(\dfrac{x-6}{(x-3)^2(x+1)}\)
7. \(\dfrac{6x-7}{(x-5)(x-4)^2}\)
8. \(\dfrac{2x^2+5x-3}{(x-1)(x-3)}\)
9. \(\dfrac{3x^3+x^2+2x-4}{(x-2)^2(x-1)}\)
10. \(\dfrac{2x^2+5x-3}{(x+1)(2x-3)(x-4)}\)

Solutions:

1. \(\dfrac{-2}{3(x-2)}+\dfrac 5{3(x+1)}\)
2. \(\dfrac 1{6(x+1)}+\dfrac{17}{6(x-5)}\)
3. \(\dfrac{2x+4}{x^2+3}+\dfrac 2{x-2}\)
4. \(\dfrac{33x+11}{37(x^2+4)}+\dfrac{12}{37(3x-1)}\)
5. \(\dfrac{2x+1}{2+x^2}+\dfrac 2{x-2}\)
6. \(\dfrac{-3}{4(x-3)^2}+\dfrac 7{16(x-3)}-\dfrac 7{16(x+1)}\)
7. \(\dfrac{-17}{(x-4)^2}-\dfrac{23}{x-4}+\dfrac{23}{x-5}\)
8. \(2-\dfrac 2{x-1}+\dfrac{15}{x-3}\)
9. \(3+\dfrac{28}{(x-2)^2}+\dfrac{14}{x-2}+\dfrac 2{x-1}\)
10. \(\dfrac{-6}{25(x+1)}-\dfrac{36}{25(2x-3)}+\dfrac{49}{25(x-4)}\)