Worksheet: Partial Fraction Expansion: Nonrepeated Linear Factors

In this worksheet, we will practice decomposing a rational fraction whose denominator has only nonrepeated linear factors into partial fractions.

Q1:

Find 𝐴 and 𝐵 such that 4 𝑥 2 ( 𝑥 + 3 ) ( 𝑥 2 ) = 𝐴 𝑥 + 3 + 𝐵 𝑥 2 .

  • A 𝐴 = 1 4 5 , 𝐵 = 6 5
  • B 𝐴 = 6 5 , 𝐵 = 1 4 5
  • C 𝐴 = 6 5 , 𝐵 = 1 4 5
  • D 𝐴 = 1 4 5 , 𝐵 = 6 5
  • E 𝐴 = 1 4 5 , 𝐵 = 6 5

Q2:

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

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

Q3:

Express in partial fractions.

  • A 1 2 ( 𝑥 + 1 ) + 2 5 ( 𝑥 + 2 ) + 7 2 0 ( 𝑥 3 )
  • B 1 2 ( 𝑥 + 1 ) + 1 5 ( 𝑥 + 2 ) + 7 2 0 ( 𝑥 3 )
  • C 1 4 ( 𝑥 + 1 ) + 2 ( 𝑥 + 2 ) + 7 2 0 ( 𝑥 3 )
  • D 1 4 ( 𝑥 + 1 ) + 2 5 ( 𝑥 + 2 ) + 7 2 0 ( 𝑥 3 )
  • E 1 4 ( 𝑥 + 1 ) + 2 5 ( 𝑥 + 2 ) + 1 4 ( 𝑥 3 )

Q4:

Find 𝐴 and 𝐵 such that

  • A 𝐴 = 2 5 , 𝐵 = 2 5
  • B 𝐴 = 2 5 , 𝐵 = 2 5
  • C 𝐴 = 2 5 , 𝐵 = 2 5
  • D 𝐴 = 2 5 , 𝐵 = 2 5
  • E 𝐴 = 1 5 , 𝐵 = 1 5

Q5:

The expression 2 𝑥 + 1 ( 𝑥 + 2 ) ( 𝑥 + 3 ) can be written in the form 𝐴 𝑥 + 3 + 𝐵 𝑥 + 2 . Find the values of 𝐴 and 𝐵 .

  • A 𝐴 = 5 , 𝐵 = 3
  • B 𝐴 = 3 , 𝐵 = 5
  • C 𝐴 = 5 , 𝐵 = 3
  • D 𝐴 = 5 , 𝐵 = 3
  • E 𝐴 = 5 , 𝐵 = 3

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