Worksheet: Polynomial Long Division with Remainder

In this worksheet, we will practice finding the quotient and remainder when polynomials are divided, including the case when the divisor is irreducible.

Q1:

Use polynomial division to simplify 2đ‘Ĩ+5đ‘Ĩ+7đ‘Ĩ+4đ‘Ĩ+1īŠŠīŠ¨.

  • Ađ‘Ĩ+5đ‘Ĩ+2īŠ¨
  • Bđ‘Ĩ+3đ‘Ĩ+4īŠ¨
  • C2đ‘Ĩ+5đ‘Ĩ+4īŠ¨
  • D2đ‘Ĩ+3đ‘Ĩ+4īŠ¨
  • E2đ‘Ĩ+5đ‘Ĩ+2īŠ¨

Q2:

Write đ‘Ĩ−2đ‘Ĩ−21đ‘Ĩ−7đ‘Ĩ+6đ‘Ĩ+3đ‘Ĩ−2īŠĒīŠŠīŠ¨īŠ¨ in the form 𝑞(đ‘Ĩ)+𝑟(đ‘Ĩ)𝑑(đ‘Ĩ) where the degree of 𝑟(đ‘Ĩ) is less than that of 𝑑(đ‘Ĩ).

  • Ađ‘Ĩ+đ‘Ĩ−20−65đ‘Ĩ−34đ‘Ĩ+3đ‘Ĩ−2īŠ¨īŠ¨
  • Bđ‘Ĩ+đ‘Ĩ−20+65đ‘Ĩ−34đ‘Ĩ+3đ‘Ĩ−2īŠ¨īŠ¨
  • Cđ‘Ĩ−5đ‘Ĩ−4−5đ‘Ĩ+2đ‘Ĩ+3đ‘Ĩ−2īŠ¨īŠ¨
  • Dđ‘Ĩ−5đ‘Ĩ−4−đ‘Ĩ+3đ‘Ĩ−25đ‘Ĩ+2īŠ¨īŠ¨
  • Eđ‘Ĩ−5đ‘Ĩ−4+5đ‘Ĩ+2đ‘Ĩ+3đ‘Ĩ−2īŠ¨īŠ¨

Q3:

Find the remainder 𝑟(đ‘Ĩ), and the quotient 𝑞(đ‘Ĩ) when 4đ‘Ĩ+2đ‘Ĩ−đ‘Ĩ−6īŠĒīŠŠ is divided by 2đ‘Ĩ−4đ‘Ĩ+1īŠ¨.

  • A𝑟(đ‘Ĩ)=30đ‘Ĩ−15,𝑞(đ‘Ĩ)=2đ‘Ĩ+5đ‘Ĩ+9īŠ¨
  • B𝑟(đ‘Ĩ)=30đ‘Ĩ−13,𝑞(đ‘Ĩ)=2đ‘Ĩ−3đ‘Ĩ+7īŠ¨
  • C𝑟(đ‘Ĩ)=−5đ‘Ĩ−5,𝑞(đ‘Ĩ)=2đ‘Ĩ+1īŠ¨
  • D𝑟(đ‘Ĩ)=38đ‘Ĩ−17,𝑞(đ‘Ĩ)=2đ‘Ĩ+5đ‘Ĩ+11īŠ¨
  • E𝑟(đ‘Ĩ)=30đ‘Ĩ−15,𝑞(đ‘Ĩ)=2đ‘Ĩ−4đ‘Ĩ+1īŠ¨

Q4:

Write đ‘Ĩ−2đ‘Ĩ−17đ‘Ĩ−3đ‘Ĩ+4đ‘Ĩ−2đ‘Ĩ+3đ‘ĨīŠĢīŠĒīŠŠīŠ¨īŠ¨ in the form 𝑞(đ‘Ĩ)+𝑟(đ‘Ĩ)𝑑(đ‘Ĩ).

  • Ađ‘Ĩ−5đ‘Ĩ−2đ‘Ĩ+3+5đ‘Ĩ+2đ‘Ĩ+3đ‘ĨīŠŠīŠ¨īŠ¨
  • Bđ‘Ĩ+đ‘Ĩ−14đ‘Ĩ+45−đ‘Ĩ+3đ‘Ĩ4đ‘Ĩ−2īŠŠīŠ¨īŠ¨
  • Cđ‘Ĩ−5đ‘Ĩ−2đ‘Ĩ+3−5đ‘Ĩ+2đ‘Ĩ+3đ‘ĨīŠŠīŠ¨īŠ¨
  • Dđ‘Ĩ+đ‘Ĩ−14đ‘Ĩ+45+4đ‘Ĩ−2đ‘Ĩ+3đ‘ĨīŠŠīŠ¨īŠ¨
  • Eđ‘Ĩ−5đ‘Ĩ−2đ‘Ĩ+3−đ‘Ĩ+3đ‘Ĩ5đ‘Ĩ+2īŠŠīŠ¨īŠ¨

Q5:

Given that đ‘Ĩ+4đ‘Ĩ−2đ‘Ĩ−3=đ‘Ĩ+7īŠ¨ with a remainder of 19, rewrite đ‘Ĩ+4đ‘Ĩ−2īŠ¨ in the form (đ‘Ĩ−𝑎)Ã—đ‘ž(đ‘Ĩ)+𝑓(𝑎).

  • Ađ‘Ĩ+4đ‘Ĩ−2=(đ‘Ĩ+2)(đ‘Ĩ−1)+19īŠ¨
  • Bđ‘Ĩ+4đ‘Ĩ−2=(đ‘Ĩ+7)(đ‘Ĩ−3)+19īŠ¨
  • Cđ‘Ĩ+4đ‘Ĩ−2=(đ‘Ĩ−7)(đ‘Ĩ+3)+19īŠ¨
  • Dđ‘Ĩ+4đ‘Ĩ−2=(đ‘Ĩ−7)(đ‘Ĩ−3)+19īŠ¨
  • Eđ‘Ĩ+4đ‘Ĩ−2=(đ‘Ĩ+7)(đ‘Ĩ+3)+19īŠ¨

Q6:

Find the remainder when 3đ‘Ĩ−2đ‘Ĩ+4đ‘Ĩ+5īŠŠīŠ¨ is divided by 3đ‘Ĩ+4.

Q7:

Find the remainder when 5đ‘Ĩ+2đ‘Ĩ−8īŠ¨ is divided by đ‘Ĩ−2.

Q8:

Write 3đ‘Ĩ+4đ‘Ĩ+5đ‘Ĩ+10đ‘Ĩ+5īŠŠīŠ¨ in the form of 𝑞(đ‘Ĩ)+𝑟(đ‘Ĩ)𝑑(đ‘Ĩ).

  • A3đ‘Ĩ−11đ‘Ĩ+60+290đ‘Ĩ+5īŠ¨
  • B3đ‘Ĩ+19đ‘Ĩ−90+460đ‘Ĩ+5īŠ¨
  • C3đ‘Ĩ+19đ‘Ĩ−90+đ‘Ĩ+5460īŠ¨
  • D3đ‘Ĩ−11đ‘Ĩ+60−đ‘Ĩ+5290īŠ¨
  • E3đ‘Ĩ−11đ‘Ĩ+60−290đ‘Ĩ+5īŠ¨

Q9:

Find the remainder when 2đ‘Ĩ+3đ‘Ĩ+2īŠ¨ is divided by đ‘Ĩ+1.

Q10:

Write 3đ‘Ĩ+4đ‘Ĩ+13đ‘Ĩ+2īŠŠīŠ¨ in the form 𝑞(đ‘Ĩ)+𝑟(đ‘Ĩ)𝑑(đ‘Ĩ).

  • A3đ‘Ĩ+10đ‘Ĩ+20+đ‘Ĩ+253īŠ¨
  • Bđ‘Ĩ−2đ‘Ĩ+4+5đ‘Ĩ+2īŠ¨
  • C3đ‘Ĩ+10đ‘Ĩ+20+53đ‘Ĩ+2īŠ¨
  • D3đ‘Ĩ−2đ‘Ĩ+4+đ‘Ĩ+25īŠ¨
  • E3đ‘Ĩ−2đ‘Ĩ+4+5đ‘Ĩ+2īŠ¨

Q11:

Find the remainder when 4đ‘Ĩ+4đ‘Ĩ+3īŠ¨ is divided by 2đ‘Ĩ−3.

Q12:

Find the remainder 𝑟(đ‘Ĩ) and the quotient 𝑞(đ‘Ĩ) when 3đ‘Ĩ+2đ‘Ĩ−3đ‘Ĩ−5īŠŠīŠ¨ is divided by đ‘Ĩ+4.

  • A𝑟(đ‘Ĩ)=đ‘Ĩ+4,𝑞(đ‘Ĩ)=3đ‘Ĩ−10đ‘Ĩ+37īŠ¨
  • B𝑟(đ‘Ĩ)=−153,𝑞(đ‘Ĩ)=3đ‘Ĩ−10đ‘Ĩ+37īŠ¨
  • C𝑟(đ‘Ĩ)=−229,𝑞(đ‘Ĩ)=3đ‘Ĩ+14đ‘Ĩ+56īŠ¨
  • D𝑟(đ‘Ĩ)=đ‘Ĩ+4,𝑞(đ‘Ĩ)=3đ‘Ĩ+14đ‘Ĩ+56īŠ¨
  • E𝑟(đ‘Ĩ)=−153,𝑞(đ‘Ĩ)=đ‘Ĩ−10đ‘Ĩ+37īŠ¨

Q13:

Find the remainder 𝑟(đ‘Ĩ), and the quotient 𝑞(đ‘Ĩ) when 2đ‘Ĩ+3đ‘Ĩ−5đ‘Ĩ−5īŠĒīŠŠ is divided by 2đ‘Ĩ−1.

  • A𝑟(đ‘Ĩ)=8,𝑞(đ‘Ĩ)=đ‘Ĩ+2đ‘Ĩ−đ‘Ĩ−3īŠŠīŠ¨
  • B𝑟(đ‘Ĩ)=34,𝑞(đ‘Ĩ)=2đ‘Ĩ+đ‘Ĩ2−3đ‘Ĩ4−174īŠŠīŠ¨
  • C𝑟(đ‘Ĩ)=−7,𝑞(đ‘Ĩ)=đ‘Ĩ+2đ‘Ĩ+đ‘Ĩ−2īŠŠīŠ¨
  • D𝑟(đ‘Ĩ)=7,𝑞(đ‘Ĩ)=đ‘Ĩ+2đ‘Ĩ−đ‘Ĩ−2īŠŠīŠ¨
  • E𝑟(đ‘Ĩ)=−8,𝑞(đ‘Ĩ)=đ‘Ĩ+2đ‘Ĩ−đ‘Ĩ−3īŠŠīŠ¨

Q14:

Write 2đ‘Ĩ−2đ‘Ĩ−5đ‘Ĩ+3īŠĒīŠ¨ in the form 𝑞(đ‘Ĩ)+𝑟(đ‘Ĩ)𝑑(đ‘Ĩ).

  • Ađ‘Ĩ−6đ‘Ĩ+16đ‘Ĩ−48+139đ‘Ĩ+3īŠŠīŠ¨
  • B2đ‘Ĩ−6đ‘Ĩ+16đ‘Ĩ−48+đ‘Ĩ+3139īŠŠīŠ¨
  • C2đ‘Ĩ+6đ‘Ĩ−20đ‘Ĩ−60+175đ‘Ĩ+3īŠŠīŠ¨
  • D2đ‘Ĩ−6đ‘Ĩ+16đ‘Ĩ−48+139đ‘Ĩ+3īŠŠīŠ¨
  • E2đ‘Ĩ+6đ‘Ĩ−20đ‘Ĩ−60+đ‘Ĩ+3175īŠŠīŠ¨

Q15:

Express the division 𝑝(đ‘Ĩ)𝑑(đ‘Ĩ)=2đ‘Ĩ−đ‘Ĩ+52đ‘Ĩ−5đ‘Ĩ+8īŠŠīŠ¨ in the form 𝑞(đ‘Ĩ)+𝑟(đ‘Ĩ)𝑑(đ‘Ĩ).

  • A2đ‘Ĩ−đ‘Ĩ+52đ‘Ĩ−5đ‘Ĩ+8=ī€ŧđ‘Ĩ+52īˆ+đ‘Ĩ−152đ‘Ĩ−5đ‘Ĩ+8īŠŠīŠ¨īŠŠīŠ¯īŠ¨īŠ¨
  • B2đ‘Ĩ−đ‘Ĩ+52đ‘Ĩ−5đ‘Ĩ+8=ī€ŧđ‘Ĩ+52īˆ+đ‘Ĩ−72đ‘Ĩ−5đ‘Ĩ+8īŠŠīŠ¨īŠ­īŠ¨īŠ¨
  • C2đ‘Ĩ−đ‘Ĩ+52đ‘Ĩ−5đ‘Ĩ+8=ī€ŧđ‘Ĩ+52īˆ+đ‘Ĩ−252đ‘Ĩ−5đ‘Ĩ+8īŠŠīŠ¨īŠĒīŠŠīŠ¨īŠ¨
  • D2đ‘Ĩ−đ‘Ĩ+52đ‘Ĩ−5đ‘Ĩ+8=ī€ŧđ‘Ĩ−52īˆ+−đ‘Ĩ+252đ‘Ĩ−5đ‘Ĩ+8īŠŠīŠ¨īŠĒīŠŠīŠ¨īŠ¨
  • E2đ‘Ĩ−đ‘Ĩ+52đ‘Ĩ−5đ‘Ĩ+8=ī€ŧđ‘Ĩ+52īˆ+đ‘Ĩ−152đ‘Ĩ−5đ‘Ĩ+8īŠŠīŠ¨īŠ­īŠ¨īŠ¨

Q16:

Use polynomial long division to find the quotient 𝑞(đ‘Ĩ) and the remainder 𝑟(đ‘Ĩ) for 𝑝(đ‘Ĩ)𝑑(đ‘Ĩ), where 𝑝(đ‘Ĩ)=đ‘Ĩ+đ‘Ĩ+đ‘Ĩ+đ‘Ĩ+đ‘Ĩ+1īŠ­īŠŦīŠĒīŠ¨ and 𝑑(đ‘Ĩ)=đ‘Ĩ+đ‘Ĩ+1īŠŠ.

  • A𝑞(đ‘Ĩ)=đ‘Ĩ+đ‘Ĩ−đ‘Ĩ−đ‘ĨīŠĒīŠŠīŠ¨ and 𝑟(đ‘Ĩ)=3đ‘Ĩ+đ‘Ĩ+1īŠ¨
  • B𝑞(đ‘Ĩ)=đ‘Ĩ+đ‘Ĩ+đ‘Ĩ−1īŠĒīŠŠīŠ¨ and 𝑟(đ‘Ĩ)=2đ‘Ĩ+2
  • C𝑞(đ‘Ĩ)=đ‘Ĩ+đ‘Ĩ−đ‘Ĩ−đ‘ĨīŠĒīŠŠīŠ¨ and 𝑟(đ‘Ĩ)=3đ‘Ĩ+2đ‘Ĩ+1īŠ¨
  • D𝑞(đ‘Ĩ)=đ‘Ĩ+đ‘Ĩ+đ‘Ĩ−đ‘Ĩ−2īŠĒīŠŠīŠ¨ and 𝑟(đ‘Ĩ)=đ‘Ĩ+4đ‘Ĩ+3īŠ¨
  • E𝑞(đ‘Ĩ)=đ‘Ĩ+đ‘Ĩ+đ‘Ĩ−đ‘Ĩ−2īŠĒīŠŠīŠ¨ and 𝑟(đ‘Ĩ)=3đ‘Ĩ+4đ‘Ĩ+3īŠ¨

Q17:

Write 6đ‘Ĩ−3đ‘Ĩ−2đ‘Ĩ+3đ‘Ĩ−52đ‘Ĩ−3đ‘Ĩ−3đ‘Ĩ−4īŠ§īŠ§īŠĒīŠŠīŠĢīŠ¨ in the form 𝑞(đ‘Ĩ)+𝑟(đ‘Ĩ)𝑑(đ‘Ĩ).

  • A3đ‘Ĩ−9đ‘Ĩ2−9đ‘Ĩ2−2đ‘Ĩ+274+24đ‘Ĩ++−−322đ‘Ĩ−3đ‘Ĩ−3đ‘Ĩ−4īŠŦīŠŠīŠ¨īŠĒīŠ­īŠ§ī—īŠ¨īŠ§īŠĢī—īŠĒīŠŠīŠ­ī—īŠĒīŠĢīŠ¨īŽĸīŽĄ
  • B3đ‘Ĩ−9đ‘Ĩ2−9đ‘Ĩ2−2đ‘Ĩ+274+2đ‘Ĩ−3đ‘Ĩ−3đ‘Ĩ−424đ‘Ĩ++−−32īŠŦīŠŠīŠ¨īŠĢīŠ¨īŠĒīŠ­īŠ§ī—īŠ¨īŠ§īŠĢī—īŠĒīŠŠīŠ­ī—īŠĒīŽĸīŽĄ
  • C3đ‘Ĩ+9đ‘Ĩ2+9đ‘Ĩ2+6đ‘Ĩ+274+24đ‘Ĩ++++222đ‘Ĩ−3đ‘Ĩ−3đ‘Ĩ−4īŠŦīŠŠīŠ¨īŠĒīŠ¯īŠĢī—īŠ¨īŠ¨īŠ¨īŠĢī—īŠĒīŠ§īŠŽīŠ¯ī—īŠĒīŠĢīŠ¨īŽĸīŽĄ
  • D3đ‘Ĩ+9đ‘Ĩ2+9đ‘Ĩ2+6đ‘Ĩ+274+2đ‘Ĩ−3đ‘Ĩ−3đ‘Ĩ−424đ‘Ĩ++++22īŠŦīŠŠīŠ¨īŠĢīŠ¨īŠĒīŠ¯īŠĢī—īŠ¨īŠ¨īŠ¨īŠĢī—īŠĒīŠ§īŠŽīŠ¯ī—īŠĒīŽĸīŽĄ
  • E3đ‘Ĩ+9đ‘Ĩ2+9đ‘Ĩ2+2đ‘Ĩ+274+++24đ‘Ĩ+−222đ‘Ĩ−3đ‘Ĩ−3đ‘Ĩ−4īŠŦīŠŠīŠ¨īŠ§īŠ­īŠ­ī—īŠĒīŠ­īŠ§ī—īŠ¨īŠ¨īŠ§īŠĻīŠ§ī—īŠĒīŠĢīŠ¨īŽŖīŽĸ

Q18:

Find the quotient 𝑞(đ‘Ĩ) and the remainder 𝑟(đ‘Ĩ) when 4đ‘Ĩ−2đ‘Ĩ−3đ‘Ĩ+4đ‘Ĩ−5īŠ­īŠŦīŠĒ is divided by 2đ‘Ĩ−2đ‘Ĩ−3īŠŠīŠ¨.

  • A𝑞(đ‘Ĩ)=2đ‘Ĩ+đ‘Ĩ+đ‘Ĩ+5đ‘Ĩ2+4īŠĒīŠŠīŠ¨ and 𝑟(đ‘Ĩ)=11đ‘Ĩ+23đ‘Ĩ2+7īŠ¨
  • B𝑞(đ‘Ĩ)=48đ‘Ĩ+71đ‘Ĩ2−40īŠ¨ and 𝑟(đ‘Ĩ)=2đ‘Ĩ−3đ‘Ĩ+6đ‘Ĩ−21đ‘Ĩ2+15īŠĒīŠŠīŠ¨
  • C𝑞(đ‘Ĩ)=2đ‘Ĩ−3đ‘Ĩ+6đ‘Ĩ−21đ‘Ĩ2+15īŠĒīŠŠīŠ¨ and 𝑟(đ‘Ĩ)=48đ‘Ĩ+71đ‘Ĩ2−40īŠ¨
  • D𝑞(đ‘Ĩ)=11đ‘Ĩ+23đ‘Ĩ2+7īŠ¨ and 𝑟(đ‘Ĩ)=2đ‘Ĩ+đ‘Ĩ+đ‘Ĩ+5đ‘Ĩ2+4īŠĒīŠŠīŠ¨
  • E𝑞(đ‘Ĩ)=2đ‘Ĩ+đ‘Ĩ+đ‘Ĩ+5đ‘Ĩ2+4īŠĒīŠŠīŠ¨ and 𝑟(đ‘Ĩ)=22đ‘Ĩ+23đ‘Ĩ+14īŠ¨

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