AP POLYCET MATHEMATICS Multiple Choice Questions (MCQs)

AP POLYCET Mathematics - Partial Fractions MCQs

1. 16. What is the partial fraction decomposition of \( \frac{1}{x(x^2 + 2)} \)?

   • a) \( \frac{A}{x} + \frac{Bx + C}{x^2 + 2} \)
   • b) \( \frac{A}{x} + \frac{B}{x^2 + 2} \)
   • c) \( \frac{A}{x^2 + 2} + \frac{B}{x} \)
   • d) \( \frac{A}{x} + \frac{C}{x^2 + 2} \)

   Answer: a) \( \frac{A}{x} + \frac{Bx + C}{x^2 + 2} \)

2. 17. What is the decomposition of \( \frac{1}{x(x^2 + 1)} \)?

   • a) \( \frac{A}{x} + \frac{Bx + C}{x^2 + 1} \)
   • b) \( \frac{A}{x} + \frac{B}{x^2 + 1} \)
   • c) \( \frac{A}{x} + \frac{C}{x^2 + 1} \)
   • d) \( \frac{A}{x} + \frac{B}{x^2 + 1} \)

   Answer: a) \( \frac{A}{x} + \frac{Bx + C}{x^2 + 1} \)

3. 18. What is the partial fraction decomposition of \( \frac{1}{(x + 1)^2(x + 2)} \)?

   • a) \( \frac{A}{x + 1} + \frac{B}{(x + 1)^2} + \frac{C}{x + 2} \)
   • b) \( \frac{A}{x + 1} + \frac{B}{x + 2} \)
   • c) \( \frac{A}{(x + 1)^2} + \frac{B}{x + 2} \)
   • d) \( \frac{A}{(x + 2)^2} + \frac{B}{x + 1} \)

   Answer: a) \( \frac{A}{x + 1} + \frac{B}{(x + 1)^2} + \frac{C}{x + 2} \)

4. 19. What is the partial fraction decomposition of \( \frac{4x}{x^2 - 2x + 1} \)?

   • a) \( \frac{A}{x - 1} + \frac{B}{x + 1} \)
   • b) \( \frac{A}{x - 1} + \frac{B}{(x - 1)^2} \)
   • c) \( \frac{A}{x^2 - 2x + 1} \)
   • d) \( \frac{A}{x + 1} + \frac{B}{x - 1} \)

   Answer: b) \( \frac{A}{x - 1} + \frac{B}{(x - 1)^2} \)

5. 20. What is the correct decomposition for \( \frac{1}{(x^2 - 4)(x + 3)} \)?

   • a) \( \frac{A}{x - 2} + \frac{B}{x + 2} + \frac{C}{x + 3} \)
   • b) \( \frac{A}{x - 2} + \frac{B}{x + 3} \)
   • c) \( \frac{A}{x - 2} + \frac{B}{x + 3} + \frac{C}{x + 4} \)
   • d) \( \frac{A}{x - 3} + \frac{B}{x + 3} \)

   Answer: a) \( \frac{A}{x - 2} + \frac{B}{x + 2} + \frac{C}{x + 3} \)

6. 21. What is the partial fraction decomposition of \( \frac{3x + 2}{x^2 + x - 2} \)?

   • a) \( \frac{A}{x - 2} + \frac{B}{x + 1} \)
   • b) \( \frac{A}{x - 1} + \frac{B}{x + 2} \)
   • c) \( \frac{A}{x + 1} + \frac{B}{x - 2} \)
   • d) \( \frac{A}{x - 2} + \frac{B}{x + 1} \)

   Answer: a) \( \frac{A}{x - 2} + \frac{B}{x + 1} \)

7. 22. Which of the following is the decomposition of \( \frac{1}{x^3 + x^2 - x - 1} \)?

   • a) \( \frac{A}{x - 1} + \frac{B}{x + 1} \)
   • b) \( \frac{A}{x - 1} + \frac{B}{x^2 + 1} \)
   • c) \( \frac{A}{x - 1} + \frac{B}{x^2 - 1} \)
   • d) \( \frac{A}{x^2 + x} + \frac{B}{x + 1} \)

   Answer: a) \( \frac{A}{x - 1} + \frac{B}{x + 1} \)

8. 23. The partial fraction decomposition of \( \frac{6}{x^2 - 1} \) is:

   • a) \( \frac{3}{x - 1} + \frac{3}{x + 1} \)
   • b) \( \frac{2}{x - 1} + \frac{4}{x + 1} \)
   • c) \( \frac{3}{x + 1} + \frac{3}{x - 1} \)
   • d) \( \frac{2}{x + 1} + \frac{4}{x - 1} \)

   Answer: a) \( \frac{3}{x - 1} + \frac{3}{x + 1} \)

9. 24. What is the partial fraction decomposition of \( \frac{5}{(x + 2)^2} \)?

   • a) \( \frac{A}{x + 2} + \frac{B}{(x + 2)^2} \)
   • b) \( \frac{A}{x + 2} + \frac{B}{x + 1} \)
   • c) \( \frac{A}{(x + 2)^2} \)
   • d) \( \frac{A}{x + 2} \)

   Answer: a) \( \frac{A}{x + 2} + \frac{B}{(x + 2)^2} \)

10. 25. What is the partial fraction decomposition of \( \frac{2x + 1}{x(x + 2)} \)?

    • a) \( \frac{A}{x} + \frac{B}{x + 2} \)
    • b) \( \frac{A}{x + 2} + \frac{B}{x} \)
    • c) \( \frac{A}{x + 2} + \frac{B}{x + 1} \)
    • d) \( \frac{A}{x} \)

    Answer: a) \( \frac{A}{x} + \frac{B}{x + 2} \)

11. 26. What is the partial fraction decomposition of \( \frac{2}{x^2 + 1} \)?

    • a) \( \frac{A}{x^2 + 1} \)
    • b) \( \frac{A}{x} + \frac{B}{x^2 + 1} \)
    • c) \( \frac{A}{x + 1} + \frac{B}{x^2 + 1} \)
    • d) \( \frac{A}{x^2 + 1} \)

    Answer: a) \( \frac{A}{x^2 + 1} \)

12. 27. Which of the following is the partial fraction decomposition of \( \frac{5x}{x^2 - 4} \)?

    • a) \( \frac{A}{x + 2} + \frac{B}{x - 2} \)
    • b) \( \frac{A}{x - 2} + \frac{B}{x + 2} \)
    • c) \( \frac{A}{x - 1} + \frac{B}{x + 2} \)
    • d) \( \frac{A}{x + 2} \)

    Answer: a) \( \frac{A}{x + 2} + \frac{B}{x - 2} \)

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