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Lesson: Using Synthetic Division to Find the Zeros of a Polynomial

Sample Question Videos

Worksheet • 2 Questions • 1 Video

Q1:

One of the zeros of the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 4 π‘₯ βˆ’ 1 7 π‘₯ + 6 0 3 2 belongs to the set { 2 , 3 , 4 } . Using synthetic division, find all the zeros of 𝑓 .

  • Aβˆ’4, 3, 5
  • B4, 2, βˆ’6
  • C4, 3, βˆ’5
  • Dβˆ’4, 2, 6
  • Eβˆ’4, 3, βˆ’5

Q2:

The function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 5 π‘₯ + 7 π‘₯ + 3 π‘₯ βˆ’ 1 0 4 3 2 has 2 real zeros and 2 imaginary zeros.

Using synthetic substitution, determine which of the values 1, 2, 3, and 4 is a root of 𝑓 ( π‘₯ ) and write 𝑓 ( π‘₯ ) in the form ( π‘₯ βˆ’ π‘Ž ) 𝑄 ( π‘₯ ) .

  • A 𝑓 ( π‘₯ ) = ( π‘₯ βˆ’ 2 ) ο€Ή π‘₯ βˆ’ 3 π‘₯ + π‘₯ + 5  3 2
  • B 𝑓 ( π‘₯ ) = ( π‘₯ βˆ’ 1 ) ο€Ή π‘₯ βˆ’ 4 π‘₯ + 3 π‘₯ + 6  3 2
  • C 𝑓 ( π‘₯ ) = ( π‘₯ + 2 ) ο€Ή π‘₯ βˆ’ 7 π‘₯ + 2 1 π‘₯ βˆ’ 3 9  3 2
  • D 𝑓 ( π‘₯ ) = ( π‘₯ + 4 ) ο€Ή π‘₯ βˆ’ 9 π‘₯ + 4 3 π‘₯ βˆ’ 1 7 5  3 2
  • E 𝑓 ( π‘₯ ) = ( π‘₯ βˆ’ 4 ) ο€Ή π‘₯ βˆ’ π‘₯ + 3 π‘₯ + 1 5  3 2

Using synthetic substitution, determine which of the values βˆ’1, βˆ’2, βˆ’3, and βˆ’4 is a root of 𝑄 ( π‘₯ ) and write 𝑄 ( π‘₯ ) in the form ( π‘₯ βˆ’ 𝑏 ) 𝑃 ( π‘₯ ) .

  • A 𝑄 ( π‘₯ ) = ( π‘₯ + 1 ) ο€Ή π‘₯ βˆ’ 4 π‘₯ + 5  2
  • B 𝑄 ( π‘₯ ) = ( π‘₯ + 1 ) ο€Ή π‘₯ βˆ’ 8 π‘₯ + 2 9  2
  • C 𝑄 ( π‘₯ ) = ( π‘₯ βˆ’ 1 ) ο€Ή π‘₯ βˆ’ 2 π‘₯ βˆ’ 1  2
  • D 𝑄 ( π‘₯ ) = ( π‘₯ βˆ’ 2 ) ο€Ή π‘₯ βˆ’ π‘₯ βˆ’ 1  2
  • E 𝑄 ( π‘₯ ) = ( π‘₯ + 2 ) ο€Ή π‘₯ βˆ’ 5 π‘₯ + 1 1  2

State all the zeros of 𝑓 .

  • A2, βˆ’1, 2 + 𝑖 , 2 βˆ’ 𝑖
  • B4, 1, 5 2 + √ 1 9 2 𝑖 , 5 2 βˆ’ √ 1 9 2 𝑖
  • Cβˆ’2, 1, 2 + 𝑖 , 2 βˆ’ 𝑖
  • D2, βˆ’1, 5 2 + √ 1 9 2 𝑖 , 5 2 βˆ’ √ 1 9 2 𝑖
  • E4, βˆ’1, 2 + 𝑖 , 2 βˆ’ 𝑖
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