Partial Fractions

Maths, HSSC-Punjab First Year. This chapter is split into 23 topics and has 566 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Irreducible quadratic factors 24 questions
2. Case IV: finding the constants 25 questions
3. Case I: denominators that must be factorized first 25 questions
4. Case I: three or more linear factors 25 questions
5. Case III: form of the partial fractions with an irreducible quadratic 22 questions
6. Procedure for resolving into partial fractions 24 questions
7. Exercise 5.2 questions (quadratic factors) 22 questions
8. Reducing improper fractions by division 24 questions
9. Case III: quartic denominators with two irreducible quadratics 24 questions
10. Case I: improper fractions with linear factors 25 questions
11. Adding partial fractions back into a single fraction 24 questions
12. Exercise 5.1 questions (linear factors) 30 questions
13. Case IV: form of the partial fractions for repeated irreducible quadratics 25 questions
14. Number of equations, checking and verifying a resolution 24 questions
15. Case I: two linear factors, finding the constants 25 questions
16. Case III: finding the constants 25 questions
17. Case II: repeated and distinct linear factors together 25 questions
18. Case I: form of the partial fractions for non-repeated linear factors 25 questions
19. Conditional equations and identities 24 questions
20. Case II: a single repeated linear factor 25 questions
21. Rational fractions: proper and improper 25 questions
22. Case II: form of the partial fractions for repeated linear factors 25 questions
23. Equality of polynomials and the coefficient theorem 24 questions

Sample questions: Partial Fractions

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. For a fraction with denominator (x² + 2x + 9)² and numerator of lower degree, the correct assumed form is

• A. (Ax + B)/(x² + 2x + 9) + (Cx + D)/(x² + 2x + 9)² + (Ex + F)/(x² + 2x + 9)³
• B. (Ax + B)/(x² + 2x + 9)²
• C. A/(x² + 2x + 9) + B/(x² + 2x + 9)²
• D. (Ax + B)/(x² + 2x + 9) + (Cx + D)/(x² + 2x + 9)²(correct answer)

Answer: D.Each distinct linear factor gets a constant over it, a repeated factor gets one term for each power up to its multiplicity, and each irreducible quadratic gets a linear numerator; so the form is (Ax + B)/(x² + 2x + 9) + (Cx + D)/(x² + 2x + 9)².

Q2. For a fraction with denominator (x² + 6)²(x − 6) and numerator of lower degree, the correct assumed form is

• A. (Ax + B)/(x² + 6) + (Cx + D)/(x² + 6)² + (Ex + F)/(x² + 6)³ + G/(x − 6) + H/(x − 6)²
• B. (Ax + B)/(x² + 6)² + C/(x − 6)
• C. (Ax + B)/(x² + 6) + (Cx + D)/(x² + 6)² + E/(x − 6)(correct answer)
• D. A/(x² + 6) + B/(x² + 6)² + C/(x − 6)

Answer: C.Each distinct linear factor gets a constant over it, a repeated factor gets one term for each power up to its multiplicity, and each irreducible quadratic gets a linear numerator; so the form is (Ax + B)/(x² + 6) + (Cx + D)/(x² + 6)² + E/(x − 6).

Q3. For a fraction with denominator (x² + 1)²(x + 2)² and numerator of lower degree, the correct assumed form is

• A. A/((x² + 1)²(x + 2)²)
• B. (Ax + B)/(x² + 1) + (Cx + D)/(x² + 1)² + (Ex + F)/(x² + 1)³ + G/(x + 2) + H/(x + 2)² + A/(x + 2)³
• C. (Ax + B)/(x² + 1) + (Cx + D)/(x² + 1)² + E/(x + 2) + F/(x + 2)²(correct answer)
• D. A/(x² + 1) + B/(x² + 1)² + C/(x + 2) + D/(x + 2)²

Answer: C.Each distinct linear factor gets a constant over it, a repeated factor gets one term for each power up to its multiplicity, and each irreducible quadratic gets a linear numerator; so the form is (Ax + B)/(x² + 1) + (Cx + D)/(x² + 1)² + E/(x + 2) + F/(x + 2)².

Q4. The partial fraction form assumed for P(x)/((x² − x + 9)²(x + 2)), where deg P is less than the degree of the denominator, is

• A. (Ax + B)/(x² − x + 9) + (Cx + D)/(x² − x + 9)² + (Ex + F)/(x + 2)
• B. (Ax + B)/(x² − x + 9) + (Cx + D)/(x² − x + 9)² + E/(x + 2)(correct answer)
• C. (Ax + B)/(x² − x + 9)² + C/(x + 2)
• D. A/((x² − x + 9)²(x + 2))

Answer: B.Each distinct linear factor gets a constant over it, a repeated factor gets one term for each power up to its multiplicity, and each irreducible quadratic gets a linear numerator; so the form is (Ax + B)/(x² − x + 9) + (Cx + D)/(x² − x + 9)² + E/(x + 2).

Q5. For a fraction with denominator (x² + 1)²(x − 6)² and numerator of lower degree, the correct assumed form is

• A. A/((x² + 1)²(x − 6)²)
• B. (Ax + B)/(x² + 1) + (Cx + D)/(x² + 1)² + (Ex + F)/(x − 6) + (Gx + H)/(x − 6)²
• C. (Ax + B)/(x² + 1) + (Cx + D)/(x² + 1)² + E/(x − 6) + F/(x − 6)²(correct answer)
• D. (Ax + B)/(x² + 1) + (Cx + D)/(x² + 1)² + (Ex + F)/(x² + 1)³ + G/(x − 6) + H/(x − 6)² + A/(x − 6)³

Answer: C.Each distinct linear factor gets a constant over it, a repeated factor gets one term for each power up to its multiplicity, and each irreducible quadratic gets a linear numerator; so the form is (Ax + B)/(x² + 1) + (Cx + D)/(x² + 1)² + E/(x − 6) + F/(x − 6)².

Q6. The partial fraction form assumed for P(x)/((x² + 7)²(x + 3)), where deg P is less than the degree of the denominator, is

• A. (Ax + B)/(x² + 7) + (Cx + D)/(x² + 7)² + E/(x − 3)
• B. A/(x² + 7) + B/(x² + 7)² + C/(x + 3)
• C. A/((x² + 7)²(x + 3))
• D. (Ax + B)/(x² + 7) + (Cx + D)/(x² + 7)² + E/(x + 3)(correct answer)

Answer: D.Each distinct linear factor gets a constant over it, a repeated factor gets one term for each power up to its multiplicity, and each irreducible quadratic gets a linear numerator; so the form is (Ax + B)/(x² + 7) + (Cx + D)/(x² + 7)² + E/(x + 3).

Q7. The partial fraction form assumed for P(x)/((x² + 1)²(x − 4)²), where deg P is less than the degree of the denominator, is

• A. (Ax + B)/(x² + 1) + (Cx + D)/(x² + 1)² + (Ex + F)/(x² + 1)³ + G/(x − 4) + H/(x − 4)² + A/(x − 4)³
• B. (Ax + B)/(x² + 1) + (Cx + D)/(x² + 1)² + E/(x + 4) + F/(x + 4)²
• C. (Ax + B)/(x² + 1) + (Cx + D)/(x² + 1)² + E/(x − 4) + F/(x − 4)²(correct answer)
• D. A/(x² + 1) + B/(x² + 1)² + C/(x − 4) + D/(x − 4)²

Answer: C.Each distinct linear factor gets a constant over it, a repeated factor gets one term for each power up to its multiplicity, and each irreducible quadratic gets a linear numerator; so the form is (Ax + B)/(x² + 1) + (Cx + D)/(x² + 1)² + E/(x − 4) + F/(x − 4)².

Q8. For a fraction with denominator (x² + 3x + 6)²(x + 6) and numerator of lower degree, the correct assumed form is

• A. (Ax + B)/(x² + 3x + 6) + (Cx + D)/(x² + 3x + 6)² + E/(x − 6)
• B. (Ax + B)/(x² + 3x + 6) + (Cx + D)/(x² + 3x + 6)² + E/(x + 6)(correct answer)
• C. (Ax + B)/(x² + 3x + 6) + (Cx + D)/(x² + 3x + 6)² + (Ex + F)/(x² + 3x + 6)³ + G/(x + 6) + H/(x + 6)²
• D. (Ax + B)/(x² + 3x + 6)² + C/(x + 6)

Answer: B.Each distinct linear factor gets a constant over it, a repeated factor gets one term for each power up to its multiplicity, and each irreducible quadratic gets a linear numerator; so the form is (Ax + B)/(x² + 3x + 6) + (Cx + D)/(x² + 3x + 6)² + E/(x + 6).

Q9. The partial fraction resolution of (5x⁴ + 40x³ − 12x² + 108x + 11)/((x² + 1)²(x + 5)²) is

• A. (2x − 1)/(x² + 1) + (4x + 4)/(x² + 1)² − 2/(x + 5) − 4/(x + 5)²
• B. (2x − 1)/(x² + 1) + (2x + 2)/(x² + 1)² − 2/(x + 5) − 5/(x + 5)²
• C. (2x − 1)/(x² + 1) + (2x + 2)/(x² + 1)² − 2/(x + 5) − 4/(x + 5)²(correct answer)
• D. (2x − 1)/(x² + 1) + (2x + 2)/(x² + 1)² − 4/(x + 5) − 2/(x + 5)²

Answer: C.Assume the proper form, multiply through by the denominator and find the constants; the identity (5x⁴ + 40x³ − 12x² + 108x + 11)/((x² + 1)²(x + 5)²) = (2x − 1)/(x² + 1) + (2x + 2)/(x² + 1)² − 2/(x + 5) − 4/(x + 5)² is verified by recombining the terms.

Q10. The partial fraction resolution of (−7x⁵ + 35x⁴ − 128x³ + 382x² − 612x + 861)/((x² + 9)²(x − 4)²) is

• A. (3x + 2)/(x² + 9) − (3x + 6)/(x² + 9)² − 4/(x − 4) − 3/(x − 4)²
• B. −(3x + 2)/(x² + 9) − (3x + 6)/(x² + 9)² − 4/(x − 4) − 3/(x − 4)²
• C. −(3x + 2)/(x² + 9) + (3x + 5)/(x² + 9)² − 4/(x − 4) − 3/(x − 4)²
• D. −(3x + 2)/(x² + 9) + (3x + 6)/(x² + 9)² − 4/(x − 4) − 3/(x − 4)²(correct answer)

Answer: D.Assume the proper form, multiply through by the denominator and find the constants; the identity (−7x⁵ + 35x⁴ − 128x³ + 382x² − 612x + 861)/((x² + 9)²(x − 4)²) = −(3x + 2)/(x² + 9) + (3x + 6)/(x² + 9)² − 4/(x − 4) − 3/(x − 4)² is verified by recombining the terms.