Worksheet: Partial Fractions: Repeated Linear Factors

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


Consider the rational expression 𝑅=5𝑥31𝑥+39(𝑥4)(𝑥+1). The following strategy reveals it as a sum of partial fractions.

What is the value of 𝑅(𝑥4) when 𝑥=4? Let us call this 𝑎.

So, 𝑅=𝑎(𝑥4)+𝑆. What is 𝑆? Give the factored and simplified form.

  • A𝑆=(𝑥2)(𝑥4)(𝑥+1)
  • B𝑆=(𝑥2)(𝑥4)(𝑥+1)
  • C𝑆=5(𝑥+2)(𝑥4)(𝑥+1)
  • D𝑆=5(𝑥2)(𝑥4)(𝑥+1)
  • E𝑆=5(𝑥2)(𝑥4)(𝑥+1)

Repeat the first step with (𝑥4)𝑆 and (𝑥+1)𝑆 to find 𝑏 and 𝑐 so that 𝑆=𝑏(𝑥4)+𝑐(𝑥+1). What, finally, is the partial fraction decomposition of 𝑅?

  • A3𝑥+12𝑥41(𝑥4)
  • B3𝑥+1+2𝑥41(𝑥4)
  • C3𝑥+1+1𝑥42(𝑥4)
  • D3𝑥+11𝑥4+2(𝑥4)
  • E3𝑥+1+2𝑥4+1(𝑥4)


Express 𝑥2(𝑥+2)(𝑥+1) in partial fractions.

  • A2𝑥+2+1(𝑥+1)
  • B1𝑥+2+2𝑥+11(𝑥+1)
  • C2𝑥+21𝑥+11(𝑥+1)
  • D2𝑥+21𝑥+11(𝑥+1)
  • E2𝑥+21(𝑥+1)


Determine the partial fraction decomposition of 𝑥+𝑥+1𝑥(𝑥3)(𝑥+1).

  • A32(𝑥+1)+116(𝑥+1)13𝑥+1348(𝑥3)
  • B32(𝑥+1)+116(𝑥+1)+13𝑥712(𝑥3)
  • C14(𝑥+1)+116(𝑥+1)13𝑥+1348(𝑥3)
  • D14(𝑥+1)+116(𝑥+1)+13𝑥+1348(𝑥3)
  • E14(𝑥+1)+116(𝑥+1)13𝑥712(𝑥3)


Find 𝐴 and 𝐵 such that 2𝑥(𝑥3)=𝐴𝑥3+𝐵(𝑥3).

  • A𝐴=6, 𝐵=2
  • B𝐴=2, 𝐵=6
  • C𝐴=2, 𝐵=6
  • D𝐴=2, 𝐵=6
  • E𝐴=2, 𝐵=6


When the expression 2𝑥+4𝑥+6(𝑥1)(𝑥+2) is expressed as a sum of partial fractions, what form will it take?

  • A𝐴𝑥1+𝐵(𝑥1)+𝐶𝑥+2
  • B𝐴(𝑥1)+𝐵𝑥+2
  • C𝐴𝑥1+𝐵𝑥+2
  • D𝐴𝑥+𝐵(𝑥1)+𝐶𝑥+2
  • E𝐴𝑥1+𝐵𝑥+𝐶(𝑥1)+𝐷𝑥+2


Determine the partial fraction decomposition of 1𝑥(𝑥1).

  • A1𝑥+1𝑥1𝑥1
  • B1𝑥1𝑥1
  • C1𝑥1𝑥1𝑥1
  • D1𝑥+1𝑥+1𝑥1
  • E1𝑥+1𝑥1

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