Lesson Worksheet: Partial Fractions: Repeated Irreducible Quadratic Factors Mathematics

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

Q1:

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

Q2:

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

Q3:

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)

Q4:

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)

Q5:

Given that 𝑥+2𝑥+3(𝑥+𝑥+1)=𝐴𝑥+𝐵𝑥+𝑥+1+𝐶𝑥+𝐷(𝑥+𝑥+1), rewrite the equation after finding 𝐴, 𝐵, 𝐶, and 𝐷.

  • A𝑥+2𝑥+3(𝑥+𝑥+1)=1𝑥𝑥+𝑥+1+4𝑥+2(𝑥+𝑥+1)
  • B𝑥+2𝑥+3(𝑥+𝑥+1)=1𝑥𝑥+𝑥+1+2𝑥+4(𝑥+𝑥+1)
  • C𝑥+2𝑥+3(𝑥+𝑥+1)=𝑥1𝑥+𝑥+1+2𝑥4(𝑥+𝑥+1)
  • D𝑥+2𝑥+3(𝑥+𝑥+1)=1𝑥𝑥+𝑥+1+2𝑥4(𝑥+𝑥+1)
  • E𝑥+2𝑥+3(𝑥+𝑥+1)=𝑥1𝑥+𝑥+1+2𝑥+4(𝑥+𝑥+1)

Q6:

Decompose 𝑥+2(𝑥+𝑥) into partial fractions.

  • A1𝑥+2𝑥𝑥+2𝑥+1𝑥+2(𝑥+1)
  • B2𝑥𝑥+2(𝑥+1)
  • C1𝑥𝑥+2(𝑥+1)
  • D1𝑥+2𝑥+𝑥+2𝑥+1+𝑥+2(𝑥+1)
  • E1𝑥+𝑥+2𝑥+1+2𝑥+1(𝑥+1)

Q7:

Which of the following are the partial fractions of the expression 𝑥𝑥+10(𝑥+5𝑥+25)?

  • A𝐴𝑥+𝐵𝑥+5𝑥+25+𝐶𝑥+𝐷(𝑥+5𝑥+25)+𝐸𝑥+𝐹(𝑥+5𝑥+25)
  • B𝐴𝑥+𝐵𝑥+5𝑥+25+𝐶𝑥+𝐷(𝑥+5𝑥+25)
  • C𝐴𝑥+𝐵(𝑥+5)+𝐶𝑥+𝐷(𝑥+5)+𝐸𝑥+𝐹(𝑥+5)
  • D𝐴(𝑥+5)+𝐵(𝑥+5)+𝐶(𝑥+5)
  • E𝐴𝑥+5𝑥+25+𝐵(𝑥+5𝑥+25)+𝐶(𝑥+5𝑥+25)

Q8:

Express 𝑥+1𝑥+2𝑥+1 in partial fractions.

  • A𝑥1𝑥+1𝑥(𝑥+1)
  • B1𝑥+1+𝑥1(𝑥+1)
  • C𝑥1𝑥+1+1(𝑥+1)
  • D𝑥𝑥+1𝑥1(𝑥+1)
  • E𝑥𝑥+1𝑥+1(𝑥+1)

Q9:

Express 𝑥𝑥+1(𝑥+1) in partial fractions.

  • A1𝑥𝑥+1+𝑥(𝑥+1)
  • B𝑥+2𝑥+1𝑥+1(𝑥+1)
  • C𝑥+2𝑥+1+(𝑥+2)(𝑥+1)
  • D1+1𝑥+11(𝑥+1)
  • E1𝑥+2𝑥+1+(𝑥+2)(𝑥+1)

Q10:

Resolve 9𝑥+9(𝑥+4)(𝑥+1) into partial fractions.

  • A𝑥+1𝑥+4𝑥+1𝑥+1+3(𝑥+1)(𝑥+1)
  • B9𝑥+41𝑥+1+9(𝑥+4)
  • C1𝑥+41𝑥+11(𝑥+1)
  • D𝑥𝑥+4𝑥𝑥+1+3𝑥(𝑥+1)
  • E𝑥+1𝑥+4+𝑥+1𝑥+1+3(𝑥+1)(𝑥+4)

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