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.

Q1:

Express the rational expression 2𝑥−5(𝑥+2)(1−𝑥) into partial fractions.

  • A2𝑥−59(𝑥+2)+2𝑥−53(𝑥+2)+16(𝑥−1)
  • B−2𝑥−59(𝑥+2)+2𝑥+53(𝑥+2)+13(𝑥−1)−79(𝑥+1)
  • C−5+2𝑥3(𝑥+2)−2𝑥+53(𝑥+2)−13(𝑥−1)−73(𝑥+1)
  • D2𝑥−59(𝑥+2)+2𝑥−53(𝑥+2)+16(𝑥−1)−718(𝑥+1)
  • E−5+2𝑥3(𝑥+2)−2𝑥+53(𝑥+2)−13(𝑥−1)

Q2:

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

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

Q3:

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

  • A𝑥+1𝑥+1+2𝑥+1(𝑥+1)+1𝑥−1
  • B𝑥−1𝑥+1+𝑥+1(𝑥+1)+1𝑥−1
  • C𝑥+1𝑥+1+2𝑥+2(𝑥+1)+1𝑥−1
  • D𝑥+2(𝑥+1)+1𝑥−1
  • E−𝑥−1𝑥+1+2𝑥+2(𝑥+1)+1𝑥−1

Q4:

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

  • A1𝑥+2𝑥+4−2(𝑥+2𝑥+4)
  • B1𝑥+2𝑥+4+2(𝑥+2𝑥+4)
  • C𝑥+2𝑥+2𝑥+4+𝑥−1(𝑥+2𝑥+4)
  • D2𝑥+2𝑥+4−1(𝑥+2𝑥+4)
  • E−𝑥−2𝑥+2𝑥+4+1−𝑥(𝑥+2𝑥+4)

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.