Worksheet: Partial Fractions: Nonrepeated Irreducible Quadratic Factors

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In this worksheet, we will practice decomposing rational expressions into partial fractions when the denominator has nonrepeated irreducible quadratic factors.

Q1:

Express 3𝑥+1(𝑥+4)(𝑥−3) in partial fractions.

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

Q2:

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

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

Q3:

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

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

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