Worksheet: Finding Zeros of Polynomials

In this worksheet, we will practice using synthetic substitution and depressed polynomials to factor a polynomial to find all its zeros given one of them.

Q1:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 6 𝑥 + 1 4 𝑥 3 2 𝑥 4 0 4 3 2 .

Given that one zero of 𝑓 ( 𝑥 ) is 2 2 2 , find all zeros of 𝑓 ( 𝑥 ) using synthetic division.

  • A 2 2 2 , 2 + 2 2 , 1 3 𝑖 , 1 3 𝑖
  • B 2 2 2 , 2 2 2 , 1 3 𝑖 , 1 + 3 𝑖
  • C 2 2 2 , 2 2 2 , 1 3 𝑖 , 1 3 𝑖
  • D 2 2 2 , 2 + 2 2 , 1 3 𝑖 , 1 + 3 𝑖
  • E 2 2 2 , 2 + 2 2 , 1 + 3 𝑖 , 1 + 3 𝑖

Write the linear factorization of 𝑓 ( 𝑥 ) .

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

Q2:

Consider 𝑔 ( 𝑥 ) = 𝑥 + 6 𝑥 + 3 8 𝑥 + 2 4 𝑥 + 1 3 6 4 3 2 .

Given that one zero of 𝑔 ( 𝑥 ) is 3 + 5 𝑖 , find all zeros of 𝑔 ( 𝑥 ) using synthetic division.

  • A 3 + 5 𝑖 , 3 5 𝑖 , 2 , 2
  • B 3 + 5 𝑖 , 3 + 5 𝑖 , 2 𝑖 , 2 𝑖
  • C 3 + 5 𝑖 , 3 5 𝑖 , 2
  • D 3 + 5 𝑖 , 3 5 𝑖 , 2 𝑖 , 2 𝑖

Write the linear factorization of 𝑔 ( 𝑥 ) .

  • A 𝑔 ( 𝑥 ) = ( 𝑥 + 3 5 𝑖 ) ( 𝑥 + 3 + 5 𝑖 ) ( 𝑥 2 ) 2
  • B 𝑔 ( 𝑥 ) = ( 𝑥 + 3 5 𝑖 ) ( 𝑥 + 3 + 5 𝑖 ) ( 𝑥 + 2 ) ( 𝑥 2 )
  • C 𝑔 ( 𝑥 ) = ( 𝑥 3 5 𝑖 ) ( 𝑥 + 3 5 𝑖 ) ( 𝑥 + 2 𝑖 ) ( 𝑥 2 𝑖 )
  • D 𝑔 ( 𝑥 ) = ( 𝑥 + 3 5 𝑖 ) ( 𝑥 + 3 + 5 𝑖 ) ( 𝑥 + 2 𝑖 ) ( 𝑥 2 𝑖 )

Q3:

Consider ( 𝑥 ) = 1 6 𝑥 8 8 𝑥 + 3 1 3 𝑥 3 4 8 𝑥 + 1 1 7 4 3 2 .

Given that one zero of multiplicity 2 of ( 𝑥 ) is 3 4 , find all zeros of ( 𝑥 ) using synthetic division.

  • A 3 4 , 2 6 𝑖 , 2 + 6 𝑖
  • B 3 4 , 2 3 𝑖 , 2 + 3 𝑖
  • C 3 4 , 2 6 𝑖 , 2 + 6 𝑖
  • D 3 4 , 2 3 𝑖 , 2 + 3 𝑖

Write the linear factorization of ( 𝑥 ) .

  • A ( 𝑥 ) = ( 4 𝑥 3 ) ( 𝑥 + 2 + 6 𝑖 ) ( 𝑥 + 2 6 𝑖 ) 2
  • B ( 𝑥 ) = ( 4 𝑥 3 ) ( 𝑥 2 + 6 𝑖 ) ( 𝑥 2 6 𝑖 ) 2
  • C ( 𝑥 ) = ( 4 𝑥 3 ) ( 𝑥 + 2 + 3 𝑖 ) ( 𝑥 + 2 3 𝑖 ) 2
  • D ( 𝑥 ) = ( 4 𝑥 3 ) ( 𝑥 2 + 3 𝑖 ) ( 𝑥 2 3 𝑖 ) 2

Q4:

Consider the function 𝑘 ( 𝑥 ) = 5 𝑥 + 2 𝑥 3 0 𝑥 8 8 𝑥 + 4 0 4 3 2 .

Given that one zero of 𝑘 ( 𝑥 ) is 1 3 𝑖 , find all zeros of 𝑘 ( 𝑥 ) using synthetic division.

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

Write the linear factorization of 𝑘 ( 𝑥 ) .

  • A 𝑘 ( 𝑥 ) = ( 𝑥 1 + 3 𝑖 ) ( 𝑥 1 3 𝑖 ) ( 5 𝑥 2 ) ( 𝑥 + 2 )
  • B 𝑘 ( 𝑥 ) = ( 𝑥 1 + 3 𝑖 ) ( 𝑥 1 3 𝑖 ) ( 5 𝑥 1 ) ( 𝑥 + 4 )
  • C 𝑘 ( 𝑥 ) = ( 𝑥 1 + 3 𝑖 ) ( 𝑥 1 3 𝑖 ) ( 5 𝑥 + 1 ) ( 𝑥 4 )
  • D 𝑘 ( 𝑥 ) = ( 𝑥 1 + 3 𝑖 ) ( 𝑥 1 3 𝑖 ) ( 5 𝑥 + 2 ) ( 𝑥 2 )
  • E 𝑘 ( 𝑥 ) = ( 𝑥 1 + 3 𝑖 ) ( 𝑥 1 3 𝑖 ) ( 5 𝑥 + 4 ) ( 𝑥 1 )

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