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.

  • A𝑥+5𝑥+2
  • B𝑥+3𝑥+4
  • C2𝑥+5𝑥+4
  • D2𝑥+3𝑥+4
  • E2𝑥+5𝑥+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).

  • A𝑥+𝑥2065𝑥34𝑥+3𝑥2
  • B𝑥+𝑥20+65𝑥34𝑥+3𝑥2
  • C𝑥5𝑥45𝑥+2𝑥+3𝑥2
  • D𝑥5𝑥4𝑥+3𝑥25𝑥+2
  • E𝑥5𝑥4+5𝑥+2𝑥+3𝑥2

Q3:

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

  • A𝑟(𝑥)=30𝑥15,𝑞(𝑥)=2𝑥+5𝑥+9
  • B𝑟(𝑥)=30𝑥13,𝑞(𝑥)=2𝑥3𝑥+7
  • C𝑟(𝑥)=5𝑥5,𝑞(𝑥)=2𝑥+1
  • D𝑟(𝑥)=38𝑥17,𝑞(𝑥)=2𝑥+5𝑥+11
  • E𝑟(𝑥)=30𝑥15,𝑞(𝑥)=2𝑥4𝑥+1

Q4:

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

  • A𝑥5𝑥2𝑥+3+5𝑥+2𝑥+3𝑥
  • B𝑥+𝑥14𝑥+45𝑥+3𝑥4𝑥2
  • C𝑥5𝑥2𝑥+35𝑥+2𝑥+3𝑥
  • D𝑥+𝑥14𝑥+45+4𝑥2𝑥+3𝑥
  • E𝑥5𝑥2𝑥+3𝑥+3𝑥5𝑥+2

Q5:

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

  • A𝑥+4𝑥2=(𝑥+2)(𝑥1)+19
  • B𝑥+4𝑥2=(𝑥+7)(𝑥3)+19
  • C𝑥+4𝑥2=(𝑥7)(𝑥+3)+19
  • D𝑥+4𝑥2=(𝑥7)(𝑥3)+19
  • E𝑥+4𝑥2=(𝑥+7)(𝑥+3)+19

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).

  • A3𝑥11𝑥+60+290𝑥+5
  • B3𝑥+19𝑥90+460𝑥+5
  • C3𝑥+19𝑥90+𝑥+5460
  • D3𝑥11𝑥+60𝑥+5290
  • E3𝑥11𝑥+60290𝑥+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).

  • A3𝑥+10𝑥+20+𝑥+253
  • B𝑥2𝑥+4+5𝑥+2
  • C3𝑥+10𝑥+20+53𝑥+2
  • D3𝑥2𝑥+4+𝑥+25
  • E3𝑥2𝑥+4+5𝑥+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.

  • A𝑟(𝑥)=𝑥+4,𝑞(𝑥)=3𝑥10𝑥+37
  • B𝑟(𝑥)=153,𝑞(𝑥)=3𝑥10𝑥+37
  • C𝑟(𝑥)=229,𝑞(𝑥)=3𝑥+14𝑥+56
  • D𝑟(𝑥)=𝑥+4,𝑞(𝑥)=3𝑥+14𝑥+56
  • E𝑟(𝑥)=153,𝑞(𝑥)=𝑥10𝑥+37

Q13:

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

  • A𝑟(𝑥)=8,𝑞(𝑥)=𝑥+2𝑥𝑥3
  • B𝑟(𝑥)=34,𝑞(𝑥)=2𝑥+𝑥23𝑥4174
  • C𝑟(𝑥)=7,𝑞(𝑥)=𝑥+2𝑥+𝑥2
  • D𝑟(𝑥)=7,𝑞(𝑥)=𝑥+2𝑥𝑥2
  • E𝑟(𝑥)=8,𝑞(𝑥)=𝑥+2𝑥𝑥3

Q14:

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

  • A𝑥6𝑥+16𝑥48+139𝑥+3
  • B2𝑥6𝑥+16𝑥48+𝑥+3139
  • C2𝑥+6𝑥20𝑥60+175𝑥+3
  • D2𝑥6𝑥+16𝑥48+139𝑥+3
  • E2𝑥+6𝑥20𝑥60+𝑥+3175

Q15:

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

  • A2𝑥𝑥+52𝑥5𝑥+8=𝑥+52+𝑥152𝑥5𝑥+8
  • B2𝑥𝑥+52𝑥5𝑥+8=𝑥+52+𝑥72𝑥5𝑥+8
  • C2𝑥𝑥+52𝑥5𝑥+8=𝑥+52+𝑥252𝑥5𝑥+8
  • D2𝑥𝑥+52𝑥5𝑥+8=𝑥52+𝑥+252𝑥5𝑥+8
  • E2𝑥𝑥+52𝑥5𝑥+8=𝑥+52+𝑥152𝑥5𝑥+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.

  • A𝑞(𝑥)=𝑥+𝑥𝑥𝑥 and 𝑟(𝑥)=3𝑥+𝑥+1
  • B𝑞(𝑥)=𝑥+𝑥+𝑥1 and 𝑟(𝑥)=2𝑥+2
  • C𝑞(𝑥)=𝑥+𝑥𝑥𝑥 and 𝑟(𝑥)=3𝑥+2𝑥+1
  • D𝑞(𝑥)=𝑥+𝑥+𝑥𝑥2 and 𝑟(𝑥)=𝑥+4𝑥+3
  • E𝑞(𝑥)=𝑥+𝑥+𝑥𝑥2 and 𝑟(𝑥)=3𝑥+4𝑥+3

Q17:

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

  • A3𝑥9𝑥29𝑥22𝑥+274+24𝑥++322𝑥3𝑥3𝑥4
  • B3𝑥9𝑥29𝑥22𝑥+274+2𝑥3𝑥3𝑥424𝑥++32
  • C3𝑥+9𝑥2+9𝑥2+6𝑥+274+24𝑥++++222𝑥3𝑥3𝑥4
  • D3𝑥+9𝑥2+9𝑥2+6𝑥+274+2𝑥3𝑥3𝑥424𝑥++++22
  • E3𝑥+9𝑥2+9𝑥2+2𝑥+274+++24𝑥+222𝑥3𝑥3𝑥4

Q18:

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

  • A𝑞(𝑥)=2𝑥+𝑥+𝑥+5𝑥2+4 and 𝑟(𝑥)=11𝑥+23𝑥2+7
  • B𝑞(𝑥)=48𝑥+71𝑥240 and 𝑟(𝑥)=2𝑥3𝑥+6𝑥21𝑥2+15
  • C𝑞(𝑥)=2𝑥3𝑥+6𝑥21𝑥2+15 and 𝑟(𝑥)=48𝑥+71𝑥240
  • D𝑞(𝑥)=11𝑥+23𝑥2+7 and 𝑟(𝑥)=2𝑥+𝑥+𝑥+5𝑥2+4
  • E𝑞(𝑥)=2𝑥+𝑥+𝑥+5𝑥2+4 and 𝑟(𝑥)=22𝑥+23𝑥+14

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.