Worksheet: Partial Fraction Expansion: Nonrepeated Irreducible Quadratic Factors

In this worksheet, we will practice decomposing a rational fraction whose denominator has a nonrepeated irreducible quadratic factor into partial fractions.

Q1:

Express in partial fractions.

  • A 2 1 𝑥 6 7 1 6 9 ( 𝑥 + 4 ) + 1 0 1 3 ( 𝑥 3 ) 2 2
  • B 2 1 𝑥 6 7 1 6 9 ( 𝑥 + 4 ) 1 0 1 6 9 ( 𝑥 3 ) + 2 1 1 3 ( 𝑥 3 ) 2 2
  • C 2 1 𝑥 6 7 1 6 9 ( 𝑥 + 4 ) + 1 0 1 6 9 ( 𝑥 3 ) + 2 1 1 3 ( 𝑥 3 ) 2 2
  • D 2 1 𝑥 6 7 1 6 9 ( 𝑥 + 4 ) 2 1 1 6 9 ( 𝑥 3 ) + 1 0 1 3 ( 𝑥 3 ) 2 2
  • E 2 1 𝑥 6 7 1 6 9 ( 𝑥 + 4 ) + 2 1 1 6 9 ( 𝑥 3 ) + 1 0 1 3 ( 𝑥 3 ) 2 2

Q2:

The expression 3 𝑥 2 ( 𝑥 + 4 ) ( 𝑥 3 ) can be written in the form 𝐴 𝑥 + 𝐵 𝑥 + 4 + 𝐶 𝑥 3 . Find the values of 𝐴 𝐵 , , and 𝐶 .

  • A 𝐴 = 7 1 3 , 𝐵 = 1 0 1 3 , 𝐶 = 7 1 3
  • B 𝐴 = 1 1 3 , 𝐵 = 1 0 1 3 , 𝐶 = 1 1 3
  • C 𝐴 = 1 8 1 3 , 𝐵 = 7 1 3 , 𝐶 = 1 1 3
  • D 𝐴 = 7 1 3 , 𝐵 = 1 8 1 3 , 𝐶 = 7 1 3
  • E 𝐴 = 1 1 3 , 𝐵 = 1 0 1 3 , 𝐶 = 1 1 3

Q3:

Express 𝑥 3 ( 𝑥 + 2 ) ( 𝑥 1 ) 2 2 in partial fractions.

  • A 5 𝑥 + 5 3 ( 𝑥 + 2 ) 1 3 ( 𝑥 1 ) 2
  • B 2 3 ( 𝑥 + 2 ) 5 𝑥 + 5 3 ( 𝑥 1 ) 2
  • C 5 𝑥 + 1 3 ( 𝑥 + 2 ) 2 3 ( 𝑥 1 ) 2
  • D 5 𝑥 + 5 3 ( 𝑥 + 2 ) 2 3 ( 𝑥 1 ) 2
  • E 𝑥 + 1 3 ( 𝑥 + 2 ) 1 3 ( 𝑥 1 ) 2

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