Worksheet: Real and Complex Roots of Polynomials

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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 ∈ ℂ .

  • 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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