Stewart Precalc 6e Section 10.7: Partial Fractions

1 Concept - Choose Form · Level 1

For the rational function \(r\), choose from (i)–(iv) the appropriate form for its partial fraction decomposition. \(r(x) = \dfrac{4}{x (x - 2)^2}\). (i) \(\dfrac{A}{x} + \dfrac{B}{x - 2}\). (ii) \(\dfrac{A}{x} + \dfrac{B}{(x - 2)^2}\). (iii) \(\dfrac{A}{x} + \dfrac{B}{x - 2} + \dfrac{C}{(x - 2)^2}\). (iv) \(\dfrac{A}{x} + \dfrac{B}{x - 2} + \dfrac{C x + D}{(x - 2)^2}\).

2 Concept - Choose Form · Level 1

For the rational function \(r\), choose from (i)–(iv) the appropriate form for its partial fraction decomposition. \(r(x) = \dfrac{2 x + 8}{(x - 1)(x^2 + 4)}\). (i) \(\dfrac{A}{x - 1} + \dfrac{B}{x^2 + 4}\). (ii) \(\dfrac{A}{x - 1} + \dfrac{B x + C}{x^2 + 4}\). (iii) \(\dfrac{A}{x - 1} + \dfrac{B}{x + 2} + \dfrac{C}{x^2 + 4}\). (iv) \(\dfrac{A x + B}{x - 1} + \dfrac{C x + D}{x^2 + 4}\).

3 Skill - Write Form · Level 2

Write the form of the partial fraction decomposition of \(\dfrac{1}{(x - 1)(x + 2)}\). Do not determine the numerical values of the coefficients.

4 Skill - Write Form · Level 2

Write the form of the partial fraction decomposition of \(\dfrac{x}{x^2 + 3 x - 4}\). Do not determine the numerical values of the coefficients.

5 Skill - Write Form · Level 2

Write the form of the partial fraction decomposition of \(\dfrac{x^2 - 3 x + 5}{(x - 2)^2 (x + 4)}\). Do not determine the numerical values of the coefficients.

6 Skill - Write Form · Level 2

Write the form of the partial fraction decomposition of \(\dfrac{1}{x^4 - x^3}\). Do not determine the numerical values of the coefficients.

7 Skill - Write Form · Level 2

Write the form of the partial fraction decomposition of \(\dfrac{x^2}{(x - 3)(x^2 + 4)}\). Do not determine the numerical values of the coefficients.

8 Skill - Write Form · Level 2

Write the form of the partial fraction decomposition of \(\dfrac{1}{x^4 - 1}\). Do not determine the numerical values of the coefficients.

9 Skill - Write Form · Level 2

Write the form of the partial fraction decomposition of \(\dfrac{x^3 - 4 x^2 + 2}{(x^2 + 1)(x^2 + 2)}\). Do not determine the numerical values of the coefficients.

10 Skill - Write Form · Level 2

Write the form of the partial fraction decomposition of \(\dfrac{x^4 + x^2 + 1}{x^2 (x^2 + 4)^2}\). Do not determine the numerical values of the coefficients.

11 Skill - Write Form · Level 3

Write the form of the partial fraction decomposition of \(\dfrac{x^3 + x + 1}{x (2 x - 5)^3 (x^2 + 2 x + 5)^2}\). Do not determine the numerical values of the coefficients.

12 Skill - Write Form · Level 3

Write the form of the partial fraction decomposition of \(\dfrac{1}{(x^3 - 1)(x^2 - 1)}\). Do not determine the numerical values of the coefficients.

13 Skill - Find Decomposition · Level 2

Find the partial fraction decomposition of \(\dfrac{2}{(x - 1)(x + 1)}\).

14 Skill - Find Decomposition · Level 2

Find the partial fraction decomposition of \(\dfrac{2 x}{(x - 1)(x + 1)}\).

15 Skill - Find Decomposition · Level 2

Find the partial fraction decomposition of \(\dfrac{x + 6}{x (x + 3)}\).

16 Skill - Find Decomposition · Level 2

Find the partial fraction decomposition of \(\dfrac{12}{x^2 - 9}\).

17 Skill - Find Decomposition · Level 2

Find the partial fraction decomposition of \(\dfrac{x - 12}{x^2 - 4 x}\).

18 Skill - Find Decomposition · Level 2

Find the partial fraction decomposition of \(\dfrac{4}{x^2 - 4}\).

19 Skill - Find Decomposition · Level 2

Find the partial fraction decomposition of \(\dfrac{2 x + 1}{x^2 + x - 2}\).

20 Skill - Find Decomposition · Level 2

Find the partial fraction decomposition of \(\dfrac{x + 14}{x^2 - 2 x - 8}\).

21 Skill - Find Decomposition · Level 2

Find the partial fraction decomposition of \(\dfrac{8 x - 3}{2 x^2 - x}\).

22 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{x}{8 x^2 - 10 x + 3}\).

23 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{7 x - 3}{x^3 + 2 x^2 - 3 x}\).

24 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{9 x^2 - 9 x + 6}{2 x^3 - x^2 - 8 x + 4}\).

25 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{-3 x^2 - 3 x + 27}{(x + 2)(2 x^2 + 3 x - 9)}\).

26 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{x^2 + 1}{x^3 + x^2}\).

27 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{3 x^2 + 5 x - 13}{(3 x + 2)(x^2 - 4 x + 4)}\).

28 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{2 x}{4 x^2 + 12 x + 9}\).

29 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{x - 4}{(2 x - 5)^2}\).

30 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{4 x^2 - x - 2}{x^4 + 2 x^3}\).

31 Skill - Find Decomposition · Level 2

Find the partial fraction decomposition of \(\dfrac{x^3 - 2 x^2 - 4 x + 3}{x^4}\).

32 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{-10 x^2 + 27 x - 14}{(x - 1)^3 (x + 2)}\).

33 Skill - Find Decomposition · Level 4

Find the partial fraction decomposition of \(\dfrac{-2 x^2 + 5 x - 1}{x^4 - 2 x^3 + 2 x - 1}\).

34 Skill - Find Decomposition · Level 4

Find the partial fraction decomposition of \(\dfrac{3 x^3 + 22 x^2 + 53 x + 41}{(x + 2)^2 (x + 3)^2}\).

35 Skill - Find Decomposition · Level 4

Find the partial fraction decomposition of \(\dfrac{3 x^2 + 12 x - 20}{x^4 - 8 x^2 + 16}\).

36 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{x - 3}{x^3 + 3 x}\).

37 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{3 x^2 - 2 x + 8}{x^3 - x^2 + 2 x - 2}\).

38 Skill - Find Decomposition · Level 4

Find the partial fraction decomposition of \(\dfrac{2 x^3 + 7 x + 5}{(x^2 + x + 2)(x^2 + 1)}\).

39 Skill - Find Decomposition · Level 4

Find the partial fraction decomposition of \(\dfrac{x^2 + x + 1}{2 x^4 + 3 x^2 + 1}\).

40 Skill - Find Decomposition · Level 4

Find the partial fraction decomposition of \(\dfrac{x^4 + x^3 + x^2 - x + 1}{x (x^2 + 1)^2}\).

41 Skill - Find Decomposition · Level 3

Find the partial fraction decomposition of \(\dfrac{2 x^2 - x + 8}{(x^2 + 4)^2}\).

42 Skill - Find Decomposition · Level 4

Find the partial fraction decomposition of \(\dfrac{x^5 - 2 x^4 + x^3 + x + 5}{x^3 - 2 x^2 + x - 2}\).

43 Skill - Find Decomposition · Level 4

Find the partial fraction decomposition of \(\dfrac{x^5 - 3 x^4 + 3 x^3 - 4 x^2 + 4 x + 12}{(x - 2)^2 (x^2 + 2)}\).

44 Skill - Determine Constants · Level 3

Determine \(A\) and \(B\) in terms of \(a\) and \(b\). \(\dfrac{a x + b}{x^2 - 1} = \dfrac{A}{x - 1} + \dfrac{B}{x + 1}\).

45 Skill - Determine Constants · Level 3

Determine \(A\), \(B\), \(C\), and \(D\) in terms of \(a\) and \(b\). \(\dfrac{a x^3 + b x^2}{(x^2 + 1)^2} = \dfrac{A x + B}{x^2 + 1} + \dfrac{C x + D}{(x^2 + 1)^2}\).

46 Discovery - Recognize Decompositions · Level 3

Recognizing Partial Fraction Decompositions. For each expression, determine whether it is already a partial fraction decomposition or whether it can be decomposed further. (a) \(\dfrac{x}{x^2 + 1} + \dfrac{1}{x + 1}\). (b) \(\dfrac{x}{(x + 1)^2}\). (c) \(\dfrac{1}{x + 1} + \dfrac{2}{(x + 1)^2}\). (d) \(\dfrac{x + 2}{(x^2 + 1)^2}\).

47 Discovery - Assemble and Disassemble · Level 3

Assembling and Disassembling Partial Fractions. The following expression is a partial fraction decomposition: \(\dfrac{2}{x - 1} + \dfrac{1}{(x - 1)^2} + \dfrac{1}{x + 1}\). Use a common denominator to combine the terms into one fraction. Then use the techniques of this section to find its partial fraction decomposition. Did you get back the original expression?

48 Example - Distinct Linear Factors · Level 2

Find the partial fraction decomposition of \(\dfrac{5x + 7}{x^3 + 2x^2 - x - 2}\).

49 Example - Repeated Linear Factors · Level 3

Find the partial fraction decomposition of \(\dfrac{x^2 + 1}{x (x - 1)^3}\).

50 Example - Distinct Quadratic Factors · Level 3

Find the partial fraction decomposition of \(\dfrac{2 x^2 - x + 4}{x^3 + 4 x}\).

51 Example - Repeated Quadratic Factors · Level 3

Write the form of the partial fraction decomposition of \(\dfrac{x^5 - 3 x^2 + 12 x - 1}{x^3 (x^2 + x + 1)(x^2 + 2)^3}\).

52 Example - Long Division Before Partial Fractions · Level 3

Find the partial fraction decomposition of \(\dfrac{2 x^4 + 4 x^3 - 2 x^2 + x + 7}{x^3 + 2 x^2 - x - 2}\).

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