Lesson Worksheet: Partial Fractions: Nonrepeated Irreducible Quadratic Factors Mathematics • 12th Grade

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)(𝑥3) in partial fractions.

  • A21𝑥67169(𝑥+4)21169(𝑥3)+1013(𝑥3)
  • B21𝑥67169(𝑥+4)+1013(𝑥3)
  • C21𝑥67169(𝑥+4)+21169(𝑥3)+1013(𝑥3)
  • D21𝑥67169(𝑥+4)10169(𝑥3)+2113(𝑥3)
  • E21𝑥67169(𝑥+4)+10169(𝑥3)+2113(𝑥3)

Q2:

The expression 3𝑥2(𝑥+4)(𝑥3) can be written in the form 𝐴𝑥+𝐵𝑥+4+𝐶𝑥3. Find the values of 𝐴, 𝐵, and 𝐶.

  • A𝐴=113, 𝐵=1013, 𝐶=113
  • B𝐴=713, 𝐵=1813, 𝐶=713
  • C𝐴=113, 𝐵=1013, 𝐶=113
  • D𝐴=713, 𝐵=1013, 𝐶=713
  • E𝐴=1813, 𝐵=713, 𝐶=113

Q3:

Express 𝑥3(𝑥+2)(𝑥1) in partial fractions.

  • A𝑥+13(𝑥+2)13(𝑥1)
  • B23(𝑥+2)5𝑥+53(𝑥1)
  • C5𝑥+53(𝑥+2)23(𝑥1)
  • D5𝑥+13(𝑥+2)23(𝑥1)
  • E5𝑥+53(𝑥+2)13(𝑥1)

Q4:

Decompose the following to partial fractions: 𝑥+1𝑥+8.

  • A47𝑥12(𝑥+2𝑥4)+512(𝑥+2)
  • B7𝑥412(𝑥+2𝑥4)+512(𝑥+2)
  • C47𝑥12(𝑥2𝑥+4)+512(𝑥+2)
  • D7𝑥412(𝑥2𝑥+4)+512(𝑥+2)
  • E7𝑥4(𝑥2𝑥+4)+5(𝑥+2)

Q5:

Resolve the following into partial fractions: 𝑥+2𝑥+2(𝑥+1)(𝑥1)(𝑥+3)

  • A1+2𝑥(𝑥+3)+5(𝑥1)1(𝑥+1)
  • B1𝑥2(𝑥+3)+58(𝑥1)18(𝑥+1)
  • C12𝑥4(𝑥+3)18(𝑥1)+58(𝑥+1)
  • D12𝑥4(𝑥+3)+58(𝑥1)18(𝑥+1)
  • E1𝑥2(𝑥+3)18(𝑥1)+58(𝑥+1)

Q6:

Decompose the following into partial fractions: 3(𝑥+1)(𝑥+4).

  • A3𝑥+1+1𝑥+4
  • B1𝑥+1+3𝑥+4+1𝑥1+3𝑥4
  • C1𝑥+1+3𝑥+4
  • D1𝑥+11𝑥+4
  • E1𝑥+11𝑥+4+1𝑥11𝑥4

Q7:

If 𝑃(𝑥)(𝑥1)(𝑥+4)=𝐴𝑥1+5𝑥+𝐵𝑥+4, where 𝑃(𝑥) is a polynomial function, 𝑃(1)=15, and 𝑃(0)=10, find the values of the real constants 𝐴 and 𝐵.

  • A𝐴=1, 𝐵=3
  • B𝐴=2, 𝐵=1
  • C𝐴=2, 𝐵=3
  • D𝐴=3, 𝐵=1
  • E𝐴=3, 𝐵=2

Q8:

Which of the following forms can be used to find the partial fraction of the expression 2𝑥1(𝑥+1)(𝑥+4).

  • A𝐴𝑥+1+𝐵𝑥+4
  • B𝐴𝑥+1+𝐵𝑥+𝐶(𝑥+2)
  • C𝐴𝑥+1+𝐵𝑥2+𝐶𝑥+2
  • D𝐴𝑥+1+𝐵𝑥+𝐶𝑥+4
  • E𝐴𝑥+1+𝐵𝑥2+𝐶(𝑥2)

Q9:

Resolve the following into partial fractions: 4𝑥1.

  • A1𝑥11𝑥+1+1𝑥+1
  • B1𝑥1+1𝑥+1
  • C1𝑥11𝑥+12(𝑥+1)2(𝑥1)
  • D2𝑥12𝑥+1
  • E1𝑥11𝑥+12𝑥+1

Q10:

Resolve 2𝑥𝑥𝑥+𝑥1 into partial fractions.

  • A2+1𝑥11(𝑥1)1(𝑥+1)
  • B2+1𝑥1+1(𝑥1)+1𝑥+1
  • C1𝑥1+1(𝑥1)+1𝑥+1
  • D1𝑥1+𝑥1𝑥+1
  • E2+1𝑥1+𝑥1𝑥+1

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