Worksheet: Remainder and Factor Theorem with Synthetic Division

In this worksheet, we will practice identifying factors and zeros and finding the remainder of a polynomial function using the remainder and factor theorem with synthetic division.

Q1:

Liam used synthetic division to prove that 4 is a root of the polynomial 𝑓(𝑥)=2𝑥9𝑥+𝑥+12.

Using his result, factor 𝑓(𝑥) into three linear factors.

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

Q2:

Consider the function 𝑓(𝑥)=2𝑥+10𝑥+5𝑥20𝑥+3.

Using synthetic division, find the value of 𝑓(3).

State which, if any, of (𝑥3) and (𝑥+3) is a factor of 𝑓(𝑥).

  • AOnly (𝑥+3) is a factor.
  • BBoth (𝑥3) and (𝑥+3) are factors.
  • COnly (𝑥3) is a factor.
  • DNeither (𝑥3) nor (𝑥+3) is a factor.

Q3:

Consider the function 𝑓(𝑥)=2𝑥𝑥12𝑥7𝑥6.

Given that two of the three numbers 1,2, and 3 are roots of 𝑓(𝑥), use synthetic division to fully factor 𝑓(𝑥).

  • A 𝑓 ( 𝑥 ) = ( 𝑥 1 ) ( 𝑥 + 2 ) ( 2 𝑥 5 ) ( 𝑥 + 1 )
  • B 𝑓 ( 𝑥 ) = ( 𝑥 3 ) ( 𝑥 + 2 ) ( 𝑥 + 1 )
  • C 𝑓 ( 𝑥 ) = ( 𝑥 3 ) ( 𝑥 + 2 ) 𝑥 + 𝑥 + 1
  • D 𝑓 ( 𝑥 ) = ( 𝑥 3 ) ( 𝑥 + 2 ) 2 𝑥 + 𝑥 + 1
  • E 𝑓 ( 𝑥 ) = ( 𝑥 3 ) ( 𝑥 1 ) 2 𝑥 + 7 𝑥 + 1 0

Q4:

One of the zeros of the function 𝑓(𝑥)=𝑥4𝑥17𝑥+60 belongs to the set {2,3,4}. Using synthetic division, find all the zeros of 𝑓.

  • A−4, 2, 6
  • B−4, 3, −5
  • C−4, 3, 5
  • D4, 3, −5
  • E4, 2, −6

Q5:

The function 𝑓(𝑥)=𝑥5𝑥+7𝑥+3𝑥10 has 2 real zeros and 2 imaginary zeros.

Using synthetic substitution, determine which of the values 1, 2, 3, and 4 is a root of 𝑓(𝑥) and write 𝑓(𝑥) in the form (𝑥𝑎)𝑄(𝑥).

  • A 𝑓 ( 𝑥 ) = ( 𝑥 4 ) 𝑥 𝑥 + 3 𝑥 + 1 5
  • B 𝑓 ( 𝑥 ) = ( 𝑥 + 2 ) 𝑥 7 𝑥 + 2 1 𝑥 3 9
  • C 𝑓 ( 𝑥 ) = ( 𝑥 2 ) 𝑥 3 𝑥 + 𝑥 + 5
  • D 𝑓 ( 𝑥 ) = ( 𝑥 + 4 ) 𝑥 9 𝑥 + 4 3 𝑥 1 7 5
  • E 𝑓 ( 𝑥 ) = ( 𝑥 1 ) 𝑥 4 𝑥 + 3 𝑥 + 6

Using synthetic substitution, determine which of the values 1, 2, 3, and 4 is a root of 𝑄(𝑥) and write 𝑄(𝑥) in the form (𝑥𝑏)𝑃(𝑥).

  • A 𝑄 ( 𝑥 ) = ( 𝑥 + 1 ) 𝑥 4 𝑥 + 5
  • B 𝑄 ( 𝑥 ) = ( 𝑥 1 ) 𝑥 2 𝑥 1
  • C 𝑄 ( 𝑥 ) = ( 𝑥 + 1 ) 𝑥 8 𝑥 + 2 9
  • D 𝑄 ( 𝑥 ) = ( 𝑥 2 ) 𝑥 𝑥 1
  • E 𝑄 ( 𝑥 ) = ( 𝑥 + 2 ) 𝑥 5 𝑥 + 1 1

State all the zeros of 𝑓.

  • A 2 , 1, 2+𝑖, 2𝑖
  • B2, 1, 2+𝑖, 2𝑖
  • C4, 1, 2+𝑖, 2𝑖
  • D4, 1, 52+192𝑖, 52192𝑖
  • E2, 1, 52+192𝑖, 52192𝑖

Q6:

A polynomial 𝑓(𝑥) is divided by (𝑥𝑎). Given that (𝑥𝑎) is not a factor of 𝑓(𝑥), what is the remainder equal to?

  • A 𝑓 ( 𝑎 )
  • B 𝑓 ( 𝑟 )
  • C0
  • D 𝑓 ( 𝑥 )

Q7:

Given that (𝑥𝑎) is a factor of 𝑓(𝑥), what is the remainder when 𝑓(𝑥) is divided by (𝑥𝑎)?

Q8:

Consider the function 𝑓(𝑥)=𝑥9𝑥+3𝑥7𝑥+12.

Use synthetic division to find the quotient 𝑄(𝑥) and the remainder 𝑅 satisfying 𝑓(𝑥)=𝑄(𝑥)(𝑥+2)+𝑅.

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

Find 𝑓(2).

Q9:

Use synthetic substitution to find the value of 𝑓(20) given 𝑓(𝑥)=0.06𝑥0.14𝑥3.1𝑥+5.4.

Q10:

Consider the function 𝑓(𝑥)=𝑥+5𝑥8𝑥+9.

What does the remainder theorem tell us about 𝑓(3)?

  • A 𝑓 ( 3 ) is the remainder when we divide 𝑓(𝑥) by 3𝑥3.
  • B 𝑓 ( 3 ) is the remainder when we divide 𝑓(𝑥) by 𝑥3.
  • C 𝑓 ( 3 ) is the remainder when we divide 𝑓(𝑥) by 𝑥.
  • D 𝑓 ( 3 ) is the remainder when we divide 𝑓(𝑥) by 𝑥+3.

Hence, use synthetic division to find 𝑓(3).

Q11:

Find the value of 𝑎 given that 2𝑥+𝑎𝑥21𝑥36 is divisible by (𝑥+4).

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