Worksheet: The Linear Factorization Theorem in Polynomial Functions

In this worksheet, we will practice writing a polynomial function with real coefficients in standard form, given its zeros, using linear factorization and conjugate root theories.

Q1:

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

  • A 𝑓 ( 𝑥 ) = 𝑥 1 7 𝑥 + 9 3 𝑥 1 3 1 𝑥 1 0 6 4 3 2
  • B 𝑓 ( 𝑥 ) = 𝑥 + 1 7 𝑥 + 9 7 𝑥 + 1 8 7 𝑥 + 1 0 6 4 3 2
  • C 𝑓 ( 𝑥 ) = 𝑥 + 1 7 𝑥 1 3 𝑥 1 8 7 𝑥 + 1 0 6 4 3 2
  • D 𝑓 ( 𝑥 ) = 𝑥 1 7 𝑥 + 9 7 𝑥 1 8 7 𝑥 + 1 0 6 4 3 2
  • E 𝑓 ( 𝑥 ) = 𝑥 1 7 𝑥 9 𝑥 + 1 3 1 𝑥 1 0 6 4 3 2

Q2:

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

  • A 𝑓 ( 𝑥 ) = 𝑥 4 𝑥 + 8 𝑥 3 2 𝑥 + 1 6 𝑥 6 4 5 4 3 2
  • B 𝑓 ( 𝑥 ) = 𝑥 + 4 𝑥 8 𝑥 3 2 𝑥 + 1 6 𝑥 + 6 4 5 4 3 2
  • C 𝑓 ( 𝑥 ) = 𝑥 4 𝑥 8 𝑥 + 3 2 𝑥 + 1 6 𝑥 6 4 5 4 3 2
  • D 𝑓 ( 𝑥 ) = 𝑥 + 4 𝑥 + 8 𝑥 + 3 2 𝑥 + 1 6 𝑥 + 6 4 5 4 3 2

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

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