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.


Express 3π‘₯+1(π‘₯+4)(π‘₯βˆ’3) in partial fractions.

  • A21π‘₯βˆ’67169(π‘₯+4)βˆ’21169(π‘₯βˆ’3)+1013(π‘₯βˆ’3)
  • B21π‘₯βˆ’67169(π‘₯+4)+1013(π‘₯βˆ’3)
  • C21π‘₯βˆ’67169(π‘₯+4)+21169(π‘₯βˆ’3)+1013(π‘₯βˆ’3)
  • D21π‘₯βˆ’67169(π‘₯+4)βˆ’10169(π‘₯βˆ’3)+2113(π‘₯βˆ’3)
  • E21π‘₯βˆ’67169(π‘₯+4)+10169(π‘₯βˆ’3)+2113(π‘₯βˆ’3)


The expression 3π‘₯βˆ’2(π‘₯+4)(π‘₯βˆ’3) can be written in the form 𝐴π‘₯+𝐡π‘₯+4+𝐢π‘₯βˆ’3. Find the values of 𝐴, 𝐡, and 𝐢.

  • A𝐴=113, 𝐡=1013, 𝐢=113
  • B𝐴=βˆ’713, 𝐡=1813, 𝐢=713
  • C𝐴=βˆ’113, 𝐡=1013, 𝐢=113
  • D𝐴=βˆ’713, 𝐡=1013, 𝐢=713
  • E𝐴=1813, 𝐡=βˆ’713, 𝐢=113


Express π‘₯βˆ’3(π‘₯+2)(π‘₯βˆ’1) in partial fractions.

  • Aπ‘₯+13(π‘₯+2)βˆ’13(π‘₯βˆ’1)
  • B23(π‘₯+2)βˆ’5π‘₯+53(π‘₯βˆ’1)
  • C5π‘₯+53(π‘₯+2)βˆ’23(π‘₯βˆ’1)
  • D5π‘₯+13(π‘₯+2)βˆ’23(π‘₯βˆ’1)
  • E5π‘₯+53(π‘₯+2)βˆ’13(π‘₯βˆ’1)


Decompose the following to partial fractions: π‘₯+1π‘₯+8.

  • A4βˆ’7π‘₯12(π‘₯+2π‘₯βˆ’4)+512(π‘₯+2)
  • B7π‘₯βˆ’412(π‘₯+2π‘₯βˆ’4)+512(π‘₯+2)
  • C4βˆ’7π‘₯12(π‘₯βˆ’2π‘₯+4)+512(π‘₯+2)
  • D7π‘₯βˆ’412(π‘₯βˆ’2π‘₯+4)+512(π‘₯+2)
  • E7π‘₯βˆ’4(π‘₯βˆ’2π‘₯+4)+5(π‘₯+2)


Resolve the following into partial fractions: π‘₯+2π‘₯+2(π‘₯+1)(π‘₯βˆ’1)(π‘₯+3)

  • A1+2π‘₯(π‘₯+3)+5(π‘₯βˆ’1)βˆ’1(π‘₯+1)
  • B1βˆ’π‘₯2(π‘₯+3)+58(π‘₯βˆ’1)βˆ’18(π‘₯+1)
  • C1βˆ’2π‘₯4(π‘₯+3)βˆ’18(π‘₯βˆ’1)+58(π‘₯+1)
  • D1βˆ’2π‘₯4(π‘₯+3)+58(π‘₯βˆ’1)βˆ’18(π‘₯+1)
  • E1βˆ’π‘₯2(π‘₯+3)βˆ’18(π‘₯βˆ’1)+58(π‘₯+1)


Decompose the following into partial fractions: 3(π‘₯+1)(π‘₯+4).

  • A3π‘₯+1+1π‘₯+4
  • B1π‘₯+1+3π‘₯+4+1π‘₯βˆ’1+3π‘₯βˆ’4
  • C1π‘₯+1+3π‘₯+4
  • D1π‘₯+1βˆ’1π‘₯+4
  • E1π‘₯+1βˆ’1π‘₯+4+1π‘₯βˆ’1βˆ’1π‘₯βˆ’4


If 𝑃(π‘₯)(π‘₯βˆ’1)(π‘₯+4)=𝐴π‘₯βˆ’1+5π‘₯+𝐡π‘₯+4, where 𝑃(π‘₯) is a polynomial function, 𝑃(1)=15, and 𝑃(0)=10, find the values of the real constants 𝐴 and 𝐡.

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


Which of the following forms can be used to find the partial fraction of the expression 2π‘₯βˆ’1(π‘₯+1)(π‘₯+4).

  • A𝐴π‘₯+1+𝐡π‘₯+4
  • B𝐴π‘₯+1+𝐡π‘₯+𝐢(π‘₯+2)
  • C𝐴π‘₯+1+𝐡π‘₯βˆ’2+𝐢π‘₯+2
  • D𝐴π‘₯+1+𝐡π‘₯+𝐢π‘₯+4
  • E𝐴π‘₯+1+𝐡π‘₯βˆ’2+𝐢(π‘₯βˆ’2)


Resolve the following into partial fractions: 4π‘₯βˆ’1οŠͺ.

  • A1π‘₯βˆ’1βˆ’1π‘₯+1+1π‘₯+1
  • B1π‘₯βˆ’1+1π‘₯+1
  • C1π‘₯βˆ’1βˆ’1π‘₯+1βˆ’2(π‘₯+1)βˆ’2(π‘₯βˆ’1)
  • D2π‘₯βˆ’1βˆ’2π‘₯+1
  • E1π‘₯βˆ’1βˆ’1π‘₯+1βˆ’2π‘₯+1


Resolve 2π‘₯π‘₯βˆ’π‘₯+π‘₯βˆ’1 into partial fractions.

  • A2+1π‘₯βˆ’1βˆ’1(π‘₯βˆ’1)βˆ’1(π‘₯+1)
  • B2+1π‘₯βˆ’1+1(π‘₯βˆ’1)+1π‘₯+1
  • C1π‘₯βˆ’1+1(π‘₯βˆ’1)+1π‘₯+1
  • D1π‘₯βˆ’1+π‘₯βˆ’1π‘₯+1
  • E2+1π‘₯βˆ’1+π‘₯βˆ’1π‘₯+1

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