Worksheet: Factor Theorem and Synthetic Division

In this worksheet, we will practice using synthetic division and factor theorem to determine whether a binomial is a factor of a polynomial and find the remaining factors.

Q1:

Amir used synthetic division to prove that 4 is a root of the polynomial .

Using his result, factor into three linear factors.

  • A
  • B
  • C
  • D
  • E

Q2:

Consider the function 𝑓 ( 𝑥 ) = 2 𝑥 + 1 0 𝑥 + 5 𝑥 2 0 𝑥 + 3 4 3 2 .

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

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

  • ANeither ( 𝑥 3 ) nor ( 𝑥 + 3 ) is a factor.
  • BBoth ( 𝑥 3 ) and ( 𝑥 + 3 ) are factors.
  • COnly ( 𝑥 3 ) is a factor.
  • DOnly ( 𝑥 + 3 ) is a factor.

Q3:

Consider the function 𝑓 ( 𝑥 ) = 2 𝑥 𝑥 1 2 𝑥 7 𝑥 6 4 3 2 .

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

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