Worksheet: Linear Factorization and Conjugate Root Theorems

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In this worksheet, we will practice writing a polynomial function given its zeros using linear factorization and conjugate root theorems.

Q1:

Write a polynomial function of the least degree with real coefficients in standard form given that it has 1,2, and 7+2𝑖 as zeros.

  • A 𝑓 ( 𝑥 ) = 𝑥 − 1 7 𝑥 + 9 3 𝑥 − 1 3 1 𝑥 − 1 0 6
  • B 𝑓 ( 𝑥 ) = 𝑥 − 1 7 𝑥 − 9 𝑥 + 1 3 1 𝑥 − 1 0 6
  • C 𝑓 ( 𝑥 ) = 𝑥 + 1 7 𝑥 + 9 7 𝑥 + 1 8 7 𝑥 + 1 0 6
  • D 𝑓 ( 𝑥 ) = 𝑥 − 1 7 𝑥 + 9 7 𝑥 − 1 8 7 𝑥 + 1 0 6
  • E 𝑓 ( 𝑥 ) = 𝑥 + 1 7 𝑥 − 1 3 𝑥 − 1 8 7 𝑥 + 1 0 6

Q2:

Write a polynomial function of least degree with real coefficients in standard form given that it has −4 and 2𝑖 (multiplicity 2) as zeros.

  • A 𝑓 ( 𝑥 ) = 𝑥 + 4 𝑥 + 8 𝑥 + 3 2 𝑥 + 1 6 𝑥 + 6 4
  • B 𝑓 ( 𝑥 ) = 𝑥 − 4 𝑥 − 8 𝑥 + 3 2 𝑥 + 1 6 𝑥 − 6 4
  • C 𝑓 ( 𝑥 ) = 𝑥 + 4 𝑥 − 8 𝑥 − 3 2 𝑥 + 1 6 𝑥 + 6 4
  • D 𝑓 ( 𝑥 ) = 𝑥 − 4 𝑥 + 8 𝑥 − 3 2 𝑥 + 1 6 𝑥 − 6 4

Q3:

Write a polynomial function of the least degree with real coefficients in standard form given that it has √5+2, −√5+2 and 3−𝑖 as zeros.

  • A 𝑓 ( 𝑥 ) = 𝑥 − 1 0 𝑥 + 3 5 𝑥 − 4 6 𝑥 + 1 0
  • B 𝑓 ( 𝑥 ) = 𝑥 − 1 0 𝑥 + 1 3 𝑥 + 4 6 𝑥 + 1 0
  • C 𝑓 ( 𝑥 ) = 𝑥 − 1 0 𝑥 + 3 3 𝑥 − 3 4 𝑥 − 1 0
  • D 𝑓 ( 𝑥 ) = 𝑥 − 2 𝑥 − 1 5 𝑥 + 4 6 𝑥 − 1 0
  • E 𝑓 ( 𝑥 ) = 𝑥 + 1 0 𝑥 + 3 3 𝑥 + 3 4 𝑥 − 1 0

Q4:

Consider 𝑔(𝑥)=𝑥−7𝑥+11𝑥−41𝑥+180.

Write 𝑔(𝑥) as the product of linear and irreducible quadratic factors.

  • A 𝑔 ( 𝑥 ) = ( 𝑥 + 4 ) ( 𝑥 + 5 ) ( 𝑥 + 𝑥 + 4 )
  • B 𝑔 ( 𝑥 ) = ( 𝑥 + 4 ) ( 𝑥 + 5 ) ( 𝑥 + 2 𝑥 + 9 )
  • C 𝑔 ( 𝑥 ) = ( 𝑥 − 4 ) ( 𝑥 − 5 ) ( 𝑥 + 𝑥 + 4 )
  • D 𝑔 ( 𝑥 ) = ( 𝑥 − 4 ) ( 𝑥 − 5 ) ( 𝑥 + 2 𝑥 + 9 )

Write 𝑔(𝑥) as the product of linear factors.

  • A 𝑔 ( 𝑥 ) = ( 𝑥 + 4 ) ( 𝑥 + 5 ) 𝑥 + 1 − 2 √ 2 𝑖 𝑥 + 1 + 2 √ 2 𝑖
  • B 𝑔 ( 𝑥 ) = ( 𝑥 + 4 ) ( 𝑥 + 5 ) 𝑥 + 1 2 − √ 1 5 2 𝑖 𝑥 + 1 2 + √ 1 5 2 𝑖
  • C 𝑔 ( 𝑥 ) = ( 𝑥 − 4 ) ( 𝑥 − 5 ) 𝑥 + 1 − 2 √ 2 𝑖 𝑥 + 1 + 2 √ 2 𝑖
  • D 𝑔 ( 𝑥 ) = ( 𝑥 − 4 ) ( 𝑥 − 5 ) 𝑥 + 1 2 − √ 1 5 2 𝑖 𝑥 + 1 2 + √ 1 5 2 𝑖

List all zeros of 𝑔(𝑥).

  • A − 5 , − 4 , − 1 2 − √ 1 5 2 𝑖 , − 1 2 + √ 1 5 2 𝑖
  • B 4 , 5 , − 1 2 − √ 1 5 2 𝑖 , − 1 2 + √ 1 5 2 𝑖
  • C 4 , 5 , − 1 + 2 √ 2 𝑖 , − 1 − 2 √ 2 𝑖
  • D − 5 , − 4 , − 1 + 2 √ 2 𝑖 , − 1 − 2 √ 2 𝑖

Q5:

Consider ℎ(𝑥)=5𝑥−4𝑥−81𝑥+134𝑥+30.

Write ℎ(𝑥) as the product of linear and irreducible quadratic factors.

  • A ℎ ( 𝑥 ) = ( 𝑥 − 3 ) ( 5 𝑥 + 1 ) ( 𝑥 + 2 𝑥 − 1 0 )
  • B ℎ ( 𝑥 ) = ( 𝑥 + 3 ) ( 5 𝑥 − 1 ) ( 𝑥 + 1 + √ 1 1 ) ( 𝑥 + 1 − √ 1 1 )
  • C ℎ ( 𝑥 ) = ( 𝑥 − 3 ) ( 5 𝑥 + 1 ) ( 𝑥 − 1 + √ 1 1 ) ( 𝑥 − 1 − √ 1 1 )
  • D ℎ ( 𝑥 ) = ( 𝑥 − 3 ) ( 5 𝑥 + 1 ) ( 𝑥 + 1 + √ 1 1 ) ( 𝑥 + 1 − √ 1 1 )
  • E ℎ ( 𝑥 ) = ( 𝑥 + 3 ) ( 5 𝑥 − 1 ) ( 𝑥 + 2 𝑥 − 1 0 )

List all zeros of ℎ(𝑥).

  • A 3 , − 1 5
  • B − 3 , 1 5
  • C − 3 , 1 5 , − 1 − √ 1 1 , √ 1 1 − 1
  • D 3 , − 1 5 , − 1 − √ 1 1 , √ 1 1 − 1
  • E 3 , − 1 5 , 1 − √ 1 1 , 1 + √ 1 1

Q6:

Consider 𝑓(𝑥)=𝑥+3𝑥−5𝑥−3𝑥+4.

Write 𝑓(𝑥) as the product of linear and irreducible quadratic factors.

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

List all zeros of 𝑓(𝑥).

  • A − 1 , − 4 , 1
  • B − 4 , 1
  • C − 1 , 4 , 1
  • D − 4 , − 1

Q7:

Consider 𝑘(𝑥)=−3𝑥−7𝑥−7𝑥+15𝑥+50.

Write 𝑘(𝑥) as the product of linear and irreducible quadratic factors.

  • A 𝑘 ( 𝑥 ) = − ( 𝑥 − 2 ) ( 3 𝑥 + 5 ) 𝑥 + 2 𝑥 + 5
  • B 𝑘 ( 𝑥 ) = ( 𝑥 + 2 ) ( 3 𝑥 − 5 ) 𝑥 − 2 𝑥 + 5
  • C 𝑘 ( 𝑥 ) = − ( 𝑥 − 2 ) ( 3 𝑥 + 5 ) 𝑥 − 2 𝑥 + 5
  • D 𝑘 ( 𝑥 ) = ( 𝑥 + 2 ) ( 3 𝑥 − 5 ) 𝑥 + 2 𝑥 + 5
  • E 𝑘 ( 𝑥 ) = − ( 𝑥 + 2 ) ( 3 𝑥 − 5 ) 𝑥 + 2 𝑥 + 5

Write 𝑘(𝑥) as the product of linear factors.

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

List all zeros of 𝑘(𝑥).

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

Q8:

If 𝑓(𝑥) is an irreducible polynomial, with 𝑓(𝑥)∣𝑝(𝑥)𝑞(𝑥), then which of the following is true?

  • A 𝑓 ( 𝑥 ) ∣ 𝑝 ( 𝑥 ) or 𝑓(𝑥)∣𝑞(𝑥)
  • B 𝑓 ( 𝑥 ) = 𝑝 ( 𝑥 ) 𝑞 ( 𝑥 ) .
  • C 𝑓 ( 𝑥 ) is a zero divisor.
  • D 𝑓 ( 𝑥 ) = either 𝑝(𝑥) or 𝑞(𝑥).

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