Worksheet: Partial Fraction with Nonrepeated Linear Factors

In this worksheet, we will practice decomposing rational expressions into partial fractions when the denominator has no repeated factors.

Q1:

Express 𝑥 2 ( 𝑥 + 2 ) ( 𝑥 3 ) ( 𝑥 + 1 ) in partial fractions.

  • A 1 4 ( 𝑥 + 1 ) + 2 5 ( 𝑥 + 2 ) + 7 2 0 ( 𝑥 3 )
  • B 1 2 ( 𝑥 + 1 ) + 1 5 ( 𝑥 + 2 ) + 7 2 0 ( 𝑥 3 )
  • C 1 4 ( 𝑥 + 1 ) + 2 ( 𝑥 + 2 ) + 7 2 0 ( 𝑥 3 )
  • D 1 2 ( 𝑥 + 1 ) + 2 5 ( 𝑥 + 2 ) + 7 2 0 ( 𝑥 3 )
  • E 1 4 ( 𝑥 + 1 ) + 2 5 ( 𝑥 + 2 ) + 1 4 ( 𝑥 3 )

Q2:

Find 𝐴 and 𝐵 such that 4 𝑥 2 ( 𝑥 + 3 ) ( 𝑥 2 ) = 𝐴 𝑥 + 3 + 𝐵 𝑥 2 .

  • A 𝐴 = 1 4 5 , 𝐵 = 6 5
  • B 𝐴 = 6 5 , 𝐵 = 1 4 5
  • C 𝐴 = 1 4 5 , 𝐵 = 6 5
  • D 𝐴 = 6 5 , 𝐵 = 1 4 5
  • E 𝐴 = 1 4 5 , 𝐵 = 6 5

Q3:

Express 𝑥 2 𝑥 ( 𝑥 3 ) in partial fractions.

  • A 2 𝑥 + 1 ( 𝑥 3 )
  • B 1 3 𝑥 + 2 3 ( 𝑥 3 )
  • C 2 3 𝑥 + 1 3 ( 𝑥 3 )
  • D 2 𝑥 + 1 3 ( 𝑥 3 )
  • E 2 3 𝑥 + 1 ( 𝑥 3 )

Q4:

The expression 2 𝑥 + 1 ( 𝑥 + 2 ) ( 𝑥 + 3 ) can be written in the form 𝐴 𝑥 + 3 + 𝐵 𝑥 + 2 . Find the values of 𝐴 and 𝐵 .

  • A 𝐴 = 5 , 𝐵 = 3
  • B 𝐴 = 5 , 𝐵 = 3
  • C 𝐴 = 5 , 𝐵 = 3
  • D 𝐴 = 3 , 𝐵 = 5
  • E 𝐴 = 5 , 𝐵 = 3

Q5:

Find 𝐴 and 𝐵 such that 4 ( 𝑥 + 8 ) ( 𝑥 2 ) = 𝐴 𝑥 2 + 𝐵 𝑥 + 8 .

  • A 𝐴 = 1 5 , 𝐵 = 1 5
  • B 𝐴 = 2 5 , 𝐵 = 2 5
  • C 𝐴 = 2 5 , 𝐵 = 2 5
  • D 𝐴 = 2 5 , 𝐵 = 2 5
  • E 𝐴 = 2 5 , 𝐵 = 2 5

Q6:

Mason wants to convert the rational expression 6 𝑥 + 5 𝑥 4 5 𝑥 + 6 𝑥 into partial fractions.

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

  • A 6 5 + 1 1 𝑥 + 2 0 2 5 𝑥 + 3 0 𝑥
  • B 6 5 1 1 5 𝑥 + 4
  • C 6 5 2 5 𝑥 + 3 0 𝑥 1 1 𝑥 + 2 0
  • D 6 5 1 1 𝑥 2 0 2 5 𝑥 + 3 0 𝑥
  • E 6 5 1 1 𝑥 + 2 0 2 5 𝑥 + 3 0 𝑥

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

  • A 6 5 8 3 1 5 ( 5 𝑥 + 6 ) + 2 3 𝑥
  • B 6 5 + 1 7 1 5 ( 5 𝑥 + 6 ) 2 3 𝑥
  • C 6 5 1 7 1 5 ( 5 𝑥 + 6 ) + 2 3 𝑥
  • D 5 6 + 1 7 2 2 ( 2 𝑥 1 ) 8 3 3 ( 3 𝑥 + 4 )
  • E 6 5 + 1 7 1 5 ( 5 𝑥 + 6 ) + 2 3 𝑥

Q7:

Convert the rational expression 6 𝑥 2 𝑥 + 5 𝑥 + 4 𝑥 + 3 into partial fractions.

  • A 6 𝑥 + 3 2 ( 𝑥 + 1 ) + 1 7 5 2 ( 𝑥 + 3 ) 2 6
  • B 6 𝑥 3 2 ( 𝑥 + 1 ) 1 7 5 2 ( 𝑥 + 3 ) 2 6
  • C 6 𝑥 1 7 5 2 ( 𝑥 + 1 ) + 3 2 ( 𝑥 + 3 ) 2 6
  • D 6 𝑥 3 2 ( 𝑥 + 1 ) + 1 7 5 2 ( 𝑥 + 3 ) 2 6
  • E 6 𝑥 3 2 ( 𝑥 + 1 ) + 1 7 5 2 ( 𝑥 + 3 ) + 2 6

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