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 2 𝑥 + 5 𝑥 + 7 𝑥 + 4 𝑥 + 1 3 2 .

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

Q2:

Find the quotient when 2 𝑥 + 7 𝑥 8 𝑥 2 1 3 2 is divided by 2 𝑥 + 3 .

  • A 𝑥 2 𝑥 7 2
  • B 𝑥 + 2 𝑥 + 7 2
  • C 𝑥 + 7 𝑥 7 2
  • D 𝑥 + 2 𝑥 7 2
  • E 𝑥 + 7 𝑥 + 7 2

Q3:

Find the quotient when 1 6 𝑥 + 1 2 𝑥 1 4 𝑥 + 6 𝑥 4 3 2 is divided by 4 𝑥 + 6 𝑥 2 .

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

Q4:

We want to factor 3 2 𝑥 + 1 0 0 𝑥 + 3 5 𝑥 6 3 𝑥 4 3 2 into two factors. Given that one of these factors is 4 𝑥 + 9 𝑥 2 , what is the other?

  • A 8 𝑥 7 𝑥 7 2
  • B 8 𝑥 + 7 𝑥 + 7 2
  • C 8 𝑥 7 𝑥 + 7 2
  • D 8 𝑥 + 7 𝑥 7 2

Q5:

Use polynomial division to simplify 6 𝑥 + 5 𝑥 2 0 𝑥 2 1 2 𝑥 + 3 3 2 .

  • A 𝑥 2 𝑥 7 2
  • B 3 𝑥 + 2 𝑥 7 2
  • C 3 𝑥 + 5 2 𝑥 7 2
  • D 3 𝑥 2 𝑥 7 2
  • E 𝑥 + 5 2 𝑥 7 2

Q6:

Find the quotient when 𝑥 1 0 𝑥 + 1 6 𝑥 9 𝑥 4 𝑥 + 1 4 6 4 3 2 is divided by 𝑥 + 2 𝑥 7 2 .

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

Q7:

Find the quotient when 𝑥 8 𝑥 + 2 0 𝑥 2 1 4 2 is divided by 𝑥 + 2 𝑥 7 2 .

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

Q8:

Given that 2 𝑥 + 9 𝑥 2 3 𝑥 3 0 = 𝑎 𝑥 + 𝑏 𝑥 + 𝑐 ( 𝑥 + 6 ) 3 2 2 , by comparing coefficients, find 𝑎 , 𝑏 , and 𝑐 .

  • A 𝑎 = 2 , 𝑏 = 2 1 , 𝑐 = 5
  • B 𝑎 = 2 , 𝑏 = 9 , 𝑐 = 2 3
  • C 𝑎 = 2 , 𝑏 = 3 , 𝑐 = 5
  • D 𝑎 = 2 , 𝑏 = 3 , 𝑐 = 5
  • E 𝑎 = 2 , 𝑏 = 3 , 𝑐 = 3 0

Q9:

Write 𝑥 2 𝑥 2 1 𝑥 7 𝑥 + 6 𝑥 + 3 𝑥 2 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 5 𝑥 4 + 5 𝑥 + 2 𝑥 + 3 𝑥 2
  • C 𝑥 + 𝑥 2 0 6 5 𝑥 3 4 𝑥 + 3 𝑥 2
  • D 𝑥 5 𝑥 4 5 𝑥 + 2 𝑥 + 3 𝑥 2
  • E 𝑥 5 𝑥 4 𝑥 + 3 𝑥 2 5 𝑥 + 2

Q10:

The volume of a box is 1 0 𝑥 + 2 7 𝑥 + 2 𝑥 2 4 3 2 . Given that its length is 5 𝑥 4 and its width is 2 𝑥 + 3 , express the height of the box algebraically.

  • A 𝑥 + 1
  • B 𝑥 2
  • C 𝑥 1
  • D 𝑥 + 2
  • E 2 𝑥 + 1

Q11:

Find the remainder 𝑟 ( 𝑥 ) , and the quotient 𝑞 ( 𝑥 ) when 4 𝑥 + 2 𝑥 𝑥 6 4 3 is divided by 2 𝑥 4 𝑥 + 1 2 .

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

Q12:

Write 𝑥 2 𝑥 1 7 𝑥 3 𝑥 + 4 𝑥 2 𝑥 + 3 𝑥 5 4 3 2 2 in the form 𝑞 ( 𝑥 ) + 𝑟 ( 𝑥 ) 𝑑 ( 𝑥 ) .

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

Q13:

Find the quotient when 7 2 𝑥 + 5 4 𝑥 + 1 8 𝑥 4 2 6 is divided by 6 𝑥 + 2 𝑥 3 .

  • A 9 𝑥 3
  • B 7 2 𝑥 + 9 𝑥 3
  • C 9 𝑥 9 𝑥 3
  • D 9 𝑥 + 9 𝑥 3

Q14:

What must be added to 2 1 𝑥 + 7 1 𝑥 + 2 3 2 to give an expression that is divisible by 7 𝑥 + 5 ?

Q15:

What is the width of a rectangle whose area is 2 4 𝑥 7 8 𝑥 1 2 𝑥 + 1 8 𝑥 4 3 2 cm2 and whose length is 3 𝑥 + 9 𝑥 2 cm?

  • A 8 𝑥 + 2 𝑥 + 2 2 cm
  • B 8 𝑥 2 𝑥 2 2 cm
  • C 8 𝑥 + 2 𝑥 2 2 cm
  • D 8 𝑥 2 𝑥 + 2 2 cm

Q16:

The volume of a cylinder is 𝜋 2 5 𝑥 6 5 𝑥 2 9 𝑥 3 3 2 . Given that its radius is 5 𝑥 + 1 , find an expression for its height.

  • A 𝑥 1
  • B 𝑥 + 3
  • C 𝑥 + 1
  • D 𝑥 3
  • E 3 𝑥 3

Q17:

The volume of a cylinder is 𝜋 3 𝑥 + 2 4 𝑥 + 4 6 𝑥 1 6 𝑥 3 2 4 3 2 and its radius is 𝑥 + 4 . Write, in its simplest form, a polynomial for the height of the cylinder.

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

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