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Worksheet: The Linear Factorization Theorem in Polynomial Functions

Q1:

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

  • A 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 4 π‘₯ + 8 π‘₯ βˆ’ 3 2 π‘₯ + 1 6 π‘₯ βˆ’ 6 4 5 4 3 2
  • B 𝑓 ( π‘₯ ) = π‘₯ + 4 π‘₯ βˆ’ 8 π‘₯ βˆ’ 3 2 π‘₯ + 1 6 π‘₯ + 6 4 5 4 3 2
  • C 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 4 π‘₯ βˆ’ 8 π‘₯ + 3 2 π‘₯ + 1 6 π‘₯ βˆ’ 6 4 5 4 3 2
  • D 𝑓 ( π‘₯ ) = π‘₯ + 4 π‘₯ + 8 π‘₯ + 3 2 π‘₯ + 1 6 π‘₯ + 6 4 5 4 3 2

Q2:

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 ( π‘₯ ) = π‘₯ βˆ’ 1 7 π‘₯ + 9 3 π‘₯ βˆ’ 1 3 1 π‘₯ βˆ’ 1 0 6 4 3 2
  • B 𝑓 ( π‘₯ ) = π‘₯ + 1 7 π‘₯ + 9 7 π‘₯ + 1 8 7 π‘₯ + 1 0 6 4 3 2
  • C 𝑓 ( π‘₯ ) = π‘₯ + 1 7 π‘₯ βˆ’ 1 3 π‘₯ βˆ’ 1 8 7 π‘₯ + 1 0 6 4 3 2
  • D 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 7 π‘₯ + 9 7 π‘₯ βˆ’ 1 8 7 π‘₯ + 1 0 6 4 3 2
  • E 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 7 π‘₯ βˆ’ 9 π‘₯ + 1 3 1 π‘₯ βˆ’ 1 0 6 4 3 2

Q3:

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

  • A 𝑓 ( π‘₯ ) = π‘₯ + 1 0 π‘₯ + 3 3 π‘₯ + 3 4 π‘₯ βˆ’ 1 0 4 3 2
  • B 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 2 π‘₯ βˆ’ 1 5 π‘₯ + 4 6 π‘₯ βˆ’ 1 0 4 3 2
  • C 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 0 π‘₯ + 3 5 π‘₯ βˆ’ 4 6 π‘₯ + 1 0 4 3 2
  • D 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 0 π‘₯ + 3 3 π‘₯ βˆ’ 3 4 π‘₯ βˆ’ 1 0 4 3 2
  • E 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 0 π‘₯ + 1 3 π‘₯ + 4 6 π‘₯ + 1 0 4 3 2