Worksheet: Dividing Polynomials by a Divisor of First Degree with a Remainder

In this worksheet, we will practice dividing polynomials by linear divisors whose results have remainders in them.

Q1:

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

Q2:

Find the remainder 𝑟 ( 𝑥 ) and the quotient 𝑞 ( 𝑥 ) when 3 𝑥 + 2 𝑥 3 𝑥 5 3 2 is divided by 𝑥 + 4 .

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

Q3:

Find the remainder 𝑟 ( 𝑥 ) , and the quotient 𝑞 ( 𝑥 ) when 2 𝑥 + 3 𝑥 5 𝑥 5 4 3 is divided by 2 𝑥 1 .

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

Q4:

Write 3 𝑥 + 4 𝑥 + 5 𝑥 + 1 0 𝑥 + 5 3 2 in the form of 𝑞 ( 𝑥 ) + 𝑟 ( 𝑥 ) 𝑑 ( 𝑥 ) .

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

Q5:

Find the remainder when 3 𝑥 2 𝑥 + 4 𝑥 + 5 3 2 is divided by 3 𝑥 + 4 .

Q6:

Write 2 𝑥 2 𝑥 5 𝑥 + 3 4 2 in the form 𝑞 ( 𝑥 ) + 𝑟 ( 𝑥 ) 𝑑 ( 𝑥 ) .

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

Q7:

Given that 𝑥 + 4 𝑥 2 𝑥 3 = 𝑥 + 7 2 with a remainder of 19, rewrite 𝑥 + 4 𝑥 2 2 in the form ( 𝑥 𝑎 ) × 𝑞 ( 𝑥 ) + 𝑓 ( 𝑎 ) .

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

Q8:

Find the remainder when 5 𝑥 + 2 𝑥 8 2 is divided by 𝑥 2 .

Q9:

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

Q10:

Write 3 𝑥 + 4 𝑥 + 1 3 𝑥 + 2 3 2 in the form 𝑞 ( 𝑥 ) + 𝑟 ( 𝑥 ) 𝑑 ( 𝑥 ) .

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

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