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 2𝑥+5𝑥+7𝑥+4𝑥+1.

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

Q2:

Write 𝑥2𝑥21𝑥7𝑥+6𝑥+3𝑥2 in the form 𝑞(𝑥)+𝑟(𝑥)𝑑(𝑥) where the degree of 𝑟(𝑥) is less than that of 𝑑(𝑥).

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

Q3:

Find the remainder 𝑟(𝑥), and the quotient 𝑞(𝑥) when 4𝑥+2𝑥𝑥6 is divided by 2𝑥4𝑥+1.

  • A 𝑟 ( 𝑥 ) = 3 0 𝑥 1 5 , 𝑞 ( 𝑥 ) = 2 𝑥 + 5 𝑥 + 9
  • B 𝑟 ( 𝑥 ) = 3 0 𝑥 1 3 , 𝑞 ( 𝑥 ) = 2 𝑥 3 𝑥 + 7
  • C 𝑟 ( 𝑥 ) = 5 𝑥 5 , 𝑞 ( 𝑥 ) = 2 𝑥 + 1
  • D 𝑟 ( 𝑥 ) = 3 8 𝑥 1 7 , 𝑞 ( 𝑥 ) = 2 𝑥 + 5 𝑥 + 1 1
  • E 𝑟 ( 𝑥 ) = 3 0 𝑥 1 5 , 𝑞 ( 𝑥 ) = 2 𝑥 4 𝑥 + 1

Q4:

Write 𝑥2𝑥17𝑥3𝑥+4𝑥2𝑥+3𝑥 in the form 𝑞(𝑥)+𝑟(𝑥)𝑑(𝑥).

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

Q5:

Given that 𝑥+4𝑥2𝑥3=𝑥+7 with a remainder of 19, rewrite 𝑥+4𝑥2 in the form (𝑥𝑎)×𝑞(𝑥)+𝑓(𝑎).

  • A 𝑥 + 4 𝑥 2 = ( 𝑥 + 2 ) ( 𝑥 1 ) + 1 9
  • B 𝑥 + 4 𝑥 2 = ( 𝑥 + 7 ) ( 𝑥 3 ) + 1 9
  • C 𝑥 + 4 𝑥 2 = ( 𝑥 7 ) ( 𝑥 + 3 ) + 1 9
  • D 𝑥 + 4 𝑥 2 = ( 𝑥 7 ) ( 𝑥 3 ) + 1 9
  • E 𝑥 + 4 𝑥 2 = ( 𝑥 + 7 ) ( 𝑥 + 3 ) + 1 9

Q6:

Find the remainder when 3𝑥2𝑥+4𝑥+5 is divided by 3𝑥+4.

Q7:

Find the remainder when 5𝑥+2𝑥8 is divided by 𝑥2.

Q8:

Write 3𝑥+4𝑥+5𝑥+10𝑥+5 in the form of 𝑞(𝑥)+𝑟(𝑥)𝑑(𝑥).

  • A 3 𝑥 1 1 𝑥 + 6 0 + 2 9 0 𝑥 + 5
  • B 3 𝑥 + 1 9 𝑥 9 0 + 4 6 0 𝑥 + 5
  • C 3 𝑥 + 1 9 𝑥 9 0 + 𝑥 + 5 4 6 0
  • D 3 𝑥 1 1 𝑥 + 6 0 𝑥 + 5 2 9 0
  • E 3 𝑥 1 1 𝑥 + 6 0 2 9 0 𝑥 + 5

Q9:

Find the remainder when 2𝑥+3𝑥+2 is divided by 𝑥+1.

Q10:

Write 3𝑥+4𝑥+13𝑥+2 in the form 𝑞(𝑥)+𝑟(𝑥)𝑑(𝑥).

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

Q11:

Find the remainder when 4𝑥+4𝑥+3 is divided by 2𝑥3.

Q12:

Find the remainder 𝑟(𝑥) and the quotient 𝑞(𝑥) when 3𝑥+2𝑥3𝑥5 is divided by 𝑥+4.

  • A 𝑟 ( 𝑥 ) = 𝑥 + 4 , 𝑞 ( 𝑥 ) = 3 𝑥 1 0 𝑥 + 3 7
  • B 𝑟 ( 𝑥 ) = 1 5 3 , 𝑞 ( 𝑥 ) = 3 𝑥 1 0 𝑥 + 3 7
  • C 𝑟 ( 𝑥 ) = 2 2 9 , 𝑞 ( 𝑥 ) = 3 𝑥 + 1 4 𝑥 + 5 6
  • D 𝑟 ( 𝑥 ) = 𝑥 + 4 , 𝑞 ( 𝑥 ) = 3 𝑥 + 1 4 𝑥 + 5 6
  • E 𝑟 ( 𝑥 ) = 1 5 3 , 𝑞 ( 𝑥 ) = 𝑥 1 0 𝑥 + 3 7

Q13:

Find the remainder 𝑟(𝑥), and the quotient 𝑞(𝑥) when 2𝑥+3𝑥5𝑥5 is divided by 2𝑥1.

  • A 𝑟 ( 𝑥 ) = 8 , 𝑞 ( 𝑥 ) = 𝑥 + 2 𝑥 𝑥 3
  • B 𝑟 ( 𝑥 ) = 3 4 , 𝑞 ( 𝑥 ) = 2 𝑥 + 𝑥 2 3 𝑥 4 1 7 4
  • C 𝑟 ( 𝑥 ) = 7 , 𝑞 ( 𝑥 ) = 𝑥 + 2 𝑥 + 𝑥 2
  • D 𝑟 ( 𝑥 ) = 7 , 𝑞 ( 𝑥 ) = 𝑥 + 2 𝑥 𝑥 2
  • E 𝑟 ( 𝑥 ) = 8 , 𝑞 ( 𝑥 ) = 𝑥 + 2 𝑥 𝑥 3

Q14:

Write 2𝑥2𝑥5𝑥+3 in the form 𝑞(𝑥)+𝑟(𝑥)𝑑(𝑥).

  • A 𝑥 6 𝑥 + 1 6 𝑥 4 8 + 1 3 9 𝑥 + 3
  • B 2 𝑥 6 𝑥 + 1 6 𝑥 4 8 + 𝑥 + 3 1 3 9
  • C 2 𝑥 + 6 𝑥 2 0 𝑥 6 0 + 1 7 5 𝑥 + 3
  • D 2 𝑥 6 𝑥 + 1 6 𝑥 4 8 + 1 3 9 𝑥 + 3
  • E 2 𝑥 + 6 𝑥 2 0 𝑥 6 0 + 𝑥 + 3 1 7 5

Q15:

Express the division 𝑝(𝑥)𝑑(𝑥)=2𝑥𝑥+52𝑥5𝑥+8 in the form 𝑞(𝑥)+𝑟(𝑥)𝑑(𝑥).

  • A 2 𝑥 𝑥 + 5 2 𝑥 5 𝑥 + 8 = 𝑥 + 5 2 + 𝑥 1 5 2 𝑥 5 𝑥 + 8
  • B 2 𝑥 𝑥 + 5 2 𝑥 5 𝑥 + 8 = 𝑥 + 5 2 + 𝑥 7 2 𝑥 5 𝑥 + 8
  • C 2 𝑥 𝑥 + 5 2 𝑥 5 𝑥 + 8 = 𝑥 + 5 2 + 𝑥 2 5 2 𝑥 5 𝑥 + 8
  • D 2 𝑥 𝑥 + 5 2 𝑥 5 𝑥 + 8 = 𝑥 5 2 + 𝑥 + 2 5 2 𝑥 5 𝑥 + 8
  • E 2 𝑥 𝑥 + 5 2 𝑥 5 𝑥 + 8 = 𝑥 + 5 2 + 𝑥 1 5 2 𝑥 5 𝑥 + 8

Q16:

Use polynomial long division to find the quotient 𝑞(𝑥) and the remainder 𝑟(𝑥) for 𝑝(𝑥)𝑑(𝑥), where 𝑝(𝑥)=𝑥+𝑥+𝑥+𝑥+𝑥+1 and 𝑑(𝑥)=𝑥+𝑥+1.

  • A 𝑞 ( 𝑥 ) = 𝑥 + 𝑥 𝑥 𝑥 and 𝑟(𝑥)=3𝑥+𝑥+1
  • B 𝑞 ( 𝑥 ) = 𝑥 + 𝑥 + 𝑥 1 and 𝑟(𝑥)=2𝑥+2
  • C 𝑞 ( 𝑥 ) = 𝑥 + 𝑥 𝑥 𝑥 and 𝑟(𝑥)=3𝑥+2𝑥+1
  • D 𝑞 ( 𝑥 ) = 𝑥 + 𝑥 + 𝑥 𝑥 2 and 𝑟(𝑥)=𝑥+4𝑥+3
  • E 𝑞 ( 𝑥 ) = 𝑥 + 𝑥 + 𝑥 𝑥 2 and 𝑟(𝑥)=3𝑥+4𝑥+3

Q17:

Write 6𝑥3𝑥2𝑥+3𝑥52𝑥3𝑥3𝑥4 in the form 𝑞(𝑥)+𝑟(𝑥)𝑑(𝑥).

  • A 3 𝑥 9 𝑥 2 9 𝑥 2 2 𝑥 + 2 7 4 + 2 4 𝑥 + + 3 2 2 𝑥 3 𝑥 3 𝑥 4
  • B 3 𝑥 9 𝑥 2 9 𝑥 2 2 𝑥 + 2 7 4 + 2 𝑥 3 𝑥 3 𝑥 4 2 4 𝑥 + + 3 2
  • C 3 𝑥 + 9 𝑥 2 + 9 𝑥 2 + 6 𝑥 + 2 7 4 + 2 4 𝑥 + + + + 2 2 2 𝑥 3 𝑥 3 𝑥 4
  • D 3 𝑥 + 9 𝑥 2 + 9 𝑥 2 + 6 𝑥 + 2 7 4 + 2 𝑥 3 𝑥 3 𝑥 4 2 4 𝑥 + + + + 2 2
  • E 3 𝑥 + 9 𝑥 2 + 9 𝑥 2 + 2 𝑥 + 2 7 4 + + + 2 4 𝑥 + 2 2 2 𝑥 3 𝑥 3 𝑥 4

Q18:

Find the quotient 𝑞(𝑥) and the remainder 𝑟(𝑥) when 4𝑥2𝑥3𝑥+4𝑥5 is divided by 2𝑥2𝑥3.

  • A 𝑞 ( 𝑥 ) = 2 𝑥 + 𝑥 + 𝑥 + 5 𝑥 2 + 4 and 𝑟(𝑥)=11𝑥+23𝑥2+7
  • B 𝑞 ( 𝑥 ) = 4 8 𝑥 + 7 1 𝑥 2 4 0 and 𝑟(𝑥)=2𝑥3𝑥+6𝑥21𝑥2+15
  • C 𝑞 ( 𝑥 ) = 2 𝑥 3 𝑥 + 6 𝑥 2 1 𝑥 2 + 1 5 and 𝑟(𝑥)=48𝑥+71𝑥240
  • D 𝑞 ( 𝑥 ) = 1 1 𝑥 + 2 3 𝑥 2 + 7 and 𝑟(𝑥)=2𝑥+𝑥+𝑥+5𝑥2+4
  • E 𝑞 ( 𝑥 ) = 2 𝑥 + 𝑥 + 𝑥 + 5 𝑥 2 + 4 and 𝑟(𝑥)=22𝑥+23𝑥+14

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