Worksheet: Polynomial Factorization into Linear and Irreducible Quadratic Factors

In this worksheet, we will practice writing polynomial functions as the product of linear and irreducible quadratic factors with real coefficients.

Q1:

Consider 𝑓 ( 𝑥 ) = 𝑥 + 3 𝑥 5 𝑥 3 𝑥 + 4 .

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

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

Q2:

Consider 𝑔 ( 𝑥 ) = 𝑥 7 𝑥 + 1 1 𝑥 4 1 𝑥 + 1 8 0 4 3 2 .

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

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

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

  • A 𝑔 ( 𝑥 ) = ( 𝑥 + 4 ) ( 𝑥 + 5 ) 𝑥 + 1 2 1 5 2 𝑖 𝑥 + 1 2 + 1 5 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 2 𝑖 𝑥 + 1 + 2 2 𝑖

List all zeros of 𝑔 ( 𝑥 ) .

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

Q3:

Consider 𝑘 ( 𝑥 ) = 3 𝑥 7 𝑥 7 𝑥 + 1 5 𝑥 + 5 0 4 3 2 .

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

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

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

Q4:

Consider ( 𝑥 ) = 5 𝑥 4 𝑥 8 1 𝑥 + 1 3 4 𝑥 + 3 0 .

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

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

List all zeros of ( 𝑥 ) .

  • A 3 , 1 5 , 1 1 1 , 1 1 1
  • 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

Q5:

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

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

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