Worksheet: Partial Fractions: Repeated Irreducible Quadratic Factors

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


Express the rational expression 2𝑥5(𝑥+2)(1𝑥) into partial fractions.

  • A2𝑥59(𝑥+2)+2𝑥53(𝑥+2)+16(𝑥1)
  • B2𝑥59(𝑥+2)+2𝑥+53(𝑥+2)+13(𝑥1)79(𝑥+1)
  • C5+2𝑥3(𝑥+2)2𝑥+53(𝑥+2)13(𝑥1)73(𝑥+1)
  • D2𝑥59(𝑥+2)+2𝑥53(𝑥+2)+16(𝑥1)718(𝑥+1)
  • E5+2𝑥3(𝑥+2)2𝑥+53(𝑥+2)13(𝑥1)


Find 𝐴 and 𝐵 such that 𝑥+2(𝑥+𝑥+1)(𝑥1)=𝐴𝑥+𝐵𝑥+𝑥+1+𝐶𝑥+𝐷(𝑥+𝑥+1)+𝐸𝑥1.

  • A𝐴=1, 𝐵=1
  • B𝐴=23, 𝐵=13
  • C𝐴=1, 𝐵=23
  • D𝐴=13, 𝐵=23
  • E𝐴=13, 𝐵=1


Resolve 4𝑥(𝑥+1)(𝑥1) into partial fractions.

  • A𝑥+1𝑥+1+2𝑥+1(𝑥+1)+1𝑥1
  • B𝑥1𝑥+1+𝑥+1(𝑥+1)+1𝑥1
  • C𝑥+1𝑥+1+2𝑥+2(𝑥+1)+1𝑥1
  • D𝑥+2(𝑥+1)+1𝑥1
  • E𝑥1𝑥+1+2𝑥+2(𝑥+1)+1𝑥1


Express 𝑥+2𝑥+2(𝑥+2𝑥+4) in partial fractions.

  • A1𝑥+2𝑥+42(𝑥+2𝑥+4)
  • B1𝑥+2𝑥+4+2(𝑥+2𝑥+4)
  • C𝑥+2𝑥+2𝑥+4+𝑥1(𝑥+2𝑥+4)
  • D2𝑥+2𝑥+41(𝑥+2𝑥+4)
  • E𝑥2𝑥+2𝑥+4+1𝑥(𝑥+2𝑥+4)

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