Worksheet: Real and Complex Roots of Polynomials

In this worksheet, we will practice finding the number and type of roots of polynomials and finding unknown coefficients if the roots are given.

Q1:

If 𝑎 + 𝑏 𝑖 is a root of the polynomial 𝑓 ( 𝑥 ) , what is the value of 𝑓 ( 𝑎 + 𝑏 𝑖 ) ?

Q2:

Does the polynomial 𝑎 𝑧 + 𝑏 𝑧 + 𝑐 𝑧 + 𝑑 𝑧 + 𝑒 𝑧 + 𝑓 , where 𝑎 is nonzero and all the coefficients are real, have at least one real root?

  • A no
  • B cannot say
  • C yes

Q3:

If 𝑎 + 𝑏 𝑖 is a root of the equation 𝑓 ( 𝑥 ) = 0 , where 𝑓 ( 𝑥 ) is a polynomial with real coefficients, which other complex number must also be a root?

  • A 𝑎 𝑏 𝑖
  • B 𝑏 𝑎 𝑖
  • C 𝑎 𝑏 𝑖
  • D 𝑏 + 𝑎 𝑖
  • E 𝑎 + 𝑏 𝑖

Q4:

Is it possible for a polynomial with real coefficients to have exactly 3 non-real roots?

  • A yes
  • B no

Q5:

How many roots does the polynomial ( 3 𝑥 1 ) ( 𝑥 + 4 𝑥 2 ) have?

Q6:

How many real roots could the polynomial 𝑝 ( 𝑥 ) = 𝑎 𝑥 + 𝑏 𝑥 + 𝑐 𝑥 + 𝑑 𝑥 + 𝑒 𝑥 + 𝑓 have given that 𝑎 , 𝑏 , 𝑐 , 𝑑 , 𝑒 , and 𝑓 are all real?

  • A 4 or 2
  • B 5, 3, or 1
  • C only 2
  • D only 1
  • E4, 2, or 1

Q7:

Determine the type of the roots of the equation ( 𝑥 1 0 ) ( 𝑥 + 1 0 ) = 2 ( 𝑥 + 8 ) ( 𝑥 + 6 ) .

  • Areal and different
  • Bcomplex and not real
  • Creal and equal

Q8:

Determine the type of the roots of the equation 𝑥 + 4 𝑥 + 1 = 3 .

  • Acomplex and not real
  • Breal and equal
  • Creal and different

Q9:

Solve the equation 𝑥 + 1 = 0 , 𝑥 .

  • A 1 2 + 3 2 𝑖
  • B 1 2 + 3 2 𝑖 , 1 2 + 3 2 𝑖 , 1
  • C 1 2 + 3 2 𝑖 , 1 , 1 2 3 2 𝑖
  • D 1 2 + 3 2 𝑖 , 1 2 3 2 𝑖 , 1

Q10:

Given that 2 is one of the roots of the equation 𝑥 + 6 𝑥 + 2 0 = 0 , find the other two roots.

  • A 1 ± 3 𝑖
  • B 2 ± 6 𝑖
  • C 6 ± 2 𝑖
  • D 3 ± 𝑖

Q11:

Find the solution set of 𝑥 + 1 6 = 0 in the set of complex numbers.

  • A { 2 , 2 , 2 𝑖 , 2 𝑖 }
  • B { 2 , 2 , 4 𝑖 , 4 𝑖 }
  • C { 2 , 2 }
  • D { 4 , 4 𝑖 }
  • E { 2 , 2 𝑖 }

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