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

In this worksheet, we will practice decomposing rational expressions into partial fractions when the denominator has nonrepeated linear factors.


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

  • A14(π‘₯+1)+25(π‘₯+2)+720(π‘₯βˆ’3)
  • B14(π‘₯+1)+2(π‘₯+2)+720(π‘₯βˆ’3)
  • C12(π‘₯+1)+25(π‘₯+2)+720(π‘₯βˆ’3)
  • D12(π‘₯+1)+15(π‘₯+2)+720(π‘₯βˆ’3)
  • E14(π‘₯+1)+25(π‘₯+2)+14(π‘₯βˆ’3)


Find 𝐴 and 𝐡 such that 4π‘₯βˆ’2(π‘₯+3)(π‘₯βˆ’2)=𝐴π‘₯+3+𝐡π‘₯βˆ’2.

  • A𝐴=βˆ’145, 𝐡=65
  • B𝐴=βˆ’65, 𝐡=βˆ’145
  • C𝐴=145, 𝐡=65
  • D𝐴=βˆ’145, 𝐡=βˆ’65
  • E𝐴=65, 𝐡=145


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

  • A23π‘₯+1(π‘₯βˆ’3)
  • B2π‘₯+13(π‘₯βˆ’3)
  • C2π‘₯+1(π‘₯βˆ’3)
  • D13π‘₯+23(π‘₯βˆ’3)
  • E23π‘₯+13(π‘₯βˆ’3)


The expression 2π‘₯+1(π‘₯+2)(π‘₯+3) can be written in the form 𝐴π‘₯+3+𝐡π‘₯+2. Find the values of 𝐴 and 𝐡.

  • A𝐴=βˆ’5,𝐡=3
  • B𝐴=βˆ’3,𝐡=5
  • C𝐴=5,𝐡=3
  • D𝐴=βˆ’5,𝐡=βˆ’3
  • E𝐴=5,𝐡=βˆ’3


Find 𝐴 and 𝐡 such that 4(π‘₯+8)(π‘₯βˆ’2)=𝐴π‘₯βˆ’2+𝐡π‘₯+8.

  • A𝐴=25, 𝐡=25
  • B𝐴=βˆ’25, 𝐡=βˆ’25
  • C𝐴=25, 𝐡=βˆ’25
  • D𝐴=15, 𝐡=βˆ’15
  • E𝐴=βˆ’25, 𝐡=25


Mason wants to convert the rational expression 6π‘₯+5π‘₯βˆ’45π‘₯+6π‘₯ into partial fractions.

His first step is to divide the numerator by the denominator. Complete this division.

  • A65βˆ’11π‘₯βˆ’2025π‘₯+30π‘₯
  • B65+11π‘₯+2025π‘₯+30π‘₯
  • C65βˆ’11π‘₯+2025π‘₯+30π‘₯
  • D65βˆ’25π‘₯+30π‘₯11π‘₯+20
  • E65βˆ’115π‘₯+4

Mason now converts this expression into partial fractions. Convert the expression into partial fractions.

  • A65+1715(5π‘₯+6)+23π‘₯
  • B65βˆ’8315(5π‘₯+6)+23π‘₯
  • C56+1722(2π‘₯βˆ’1)βˆ’833(3π‘₯+4)
  • D65βˆ’1715(5π‘₯+6)+23π‘₯
  • E65+1715(5π‘₯+6)βˆ’23π‘₯


Convert the rational expression 6π‘₯βˆ’2π‘₯+5π‘₯+4π‘₯+3 into partial fractions.

  • A6π‘₯+32(π‘₯+1)+1752(π‘₯+3)βˆ’26
  • B6π‘₯βˆ’1752(π‘₯+1)+32(π‘₯+3)βˆ’26
  • C6π‘₯βˆ’32(π‘₯+1)+1752(π‘₯+3)+26
  • D6π‘₯βˆ’32(π‘₯+1)βˆ’1752(π‘₯+3)βˆ’26
  • E6π‘₯βˆ’32(π‘₯+1)+1752(π‘₯+3)βˆ’26


Resolve the expression 2π‘₯βˆ’4π‘₯βˆ’π‘₯βˆ’10π‘₯βˆ’5π‘₯βˆ’2π‘₯βˆ’5π‘₯+6οŠͺ into partial fractions.

  • A2π‘₯+3π‘₯βˆ’1βˆ’5π‘₯βˆ’2+1π‘₯βˆ’3
  • B3π‘₯βˆ’1+5π‘₯+2+1π‘₯βˆ’3
  • C2π‘₯+3π‘₯βˆ’1+5π‘₯+2+1π‘₯βˆ’3
  • D2π‘₯+3π‘₯+1+5π‘₯+2+1π‘₯+3
  • E3π‘₯βˆ’1+5π‘₯βˆ’2βˆ’1π‘₯βˆ’3


Which of the following is an expression for the sum of partial fractions of 2π‘₯+1(π‘₯βˆ’5)(π‘₯+1)(π‘₯βˆ’3)?

  • A𝐴π‘₯+5+𝐡π‘₯+1+𝐢π‘₯+3
  • B𝐴π‘₯βˆ’5+𝐡π‘₯+3
  • C𝐴π‘₯βˆ’5+𝐡π‘₯βˆ’1+𝐢π‘₯βˆ’3
  • D𝐴π‘₯βˆ’5+𝐡π‘₯+1+𝐢π‘₯βˆ’3
  • E𝐴π‘₯+5+𝐡π‘₯βˆ’3


Decompose 𝑃(π‘₯)(π‘₯βˆ’1)(π‘₯βˆ’3)(π‘₯βˆ’5) into partial fractions, where 𝑃(π‘₯) is a quadratic function, 𝑃(1)=8, 𝑃(3)=βˆ’8, and 𝑃(5)=24.

  • A8π‘₯βˆ’1+1π‘₯βˆ’3+24π‘₯βˆ’5
  • B1π‘₯+1βˆ’2π‘₯+3+3π‘₯+5
  • C1π‘₯βˆ’1βˆ’2π‘₯βˆ’3βˆ’3π‘₯βˆ’5
  • D8π‘₯+1βˆ’1π‘₯+3βˆ’24π‘₯+5
  • E1π‘₯βˆ’1+2π‘₯βˆ’3+3π‘₯βˆ’5

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