Worksheet: Number and Type of Roots of a Polynomial

In this worksheet, we will practice determining the possible number and types of real roots using Descartes's rule of signs and the fundamental theorem of algebra.

Q1:

Is ( 𝑥 𝑎 ) a factor of 𝑓 ( 𝑥 ) given that 𝑓 ( 𝑎 ) = 0 ?

  • Ayes
  • Bno

Q2:

Is ( 𝑥 𝑎 ) a factor of 𝑓 ( 𝑥 ) given that 𝑓 ( 𝑎 ) = 3 ?

  • Ano
  • Byes

Q3:

Find, by factoring, the zeros of the function 𝑓 ( 𝑥 ) = 1 2 𝑥 1 4 𝑥 6 2 .

  • A 1 3 , 3 2
  • B 1 3 , 3 2
  • C 2 3 , 3 4
  • D 1 3 , 3 2
  • E 2 3 , 3 4

Q4:

How many roots does the polynomial 5 𝑥 + 4 𝑥 1 3 have?

Q5:

Find, by factoring, the zeros of the function 𝑓 ( 𝑥 ) = 6 𝑥 + 3 3 𝑥 6 3 𝑥 4 3 2 .

  • A 7 , 0 , 3 2
  • B 7 , 0 , 3 2
  • C 3 , 0 , 7 2
  • D 7 , 0 , 3 2
  • E 3 , 0 , 7 2

Q6:

Which of the following lists contain all the roots of 4 𝑥 1 0 𝑥 7 6 𝑥 + 7 0 𝑥 + 3 0 0 4 3 2 ?

  • A 3 , 2 , 5 , 5 2
  • B 3 , 2 , 5 , 5 2
  • C 3 , 2 , 5 , 5 2
  • D 3 , 2 , 5 , 5 2

Q7:

Which of the following are roots of 6 𝑥 + 4 3 𝑥 1 6 9 𝑥 9 8 8 𝑥 + 1 9 4 8 𝑥 + 1 6 8 0 5 4 3 2 ?

  • A 7 , 5 2 and 2 3
  • B6, 5 2 and 2 3
  • C4, 2 5 and 2 3
  • D 6 , 5 2 and 2 3
  • E4, 6 and 7

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