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:

Determine the type of the roots of the equation ( 2 𝑥 − 4 ) + 1 7 = 0 2 .

  • Areal and different
  • Breal and equal
  • Ccomplex and not real

Q2:

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

Q3:

Does the polynomial 𝑎 𝑧 + 𝑏 𝑧 + 𝑐 𝑧 + 𝑑 𝑧 + 𝑒 𝑧 + 𝑓 5 4 3 2 , where 𝑎 is nonzero and all the coefficients are real, have at least one real root?

  • A cannot say
  • B no
  • C yes

Q4:

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 − 𝑎 + 𝑏 𝑖
  • C 𝑏 − 𝑎 𝑖
  • D 𝑎 − 𝑏 𝑖
  • E 𝑏 + 𝑎 𝑖

Q5:

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

  • A no
  • B yes

Q6:

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

Q7:

How many real roots could the polynomial 𝑝 ( 𝑥 ) = 𝑎 𝑥 + 𝑏 𝑥 + 𝑐 𝑥 + 𝑑 𝑥 + 𝑒 𝑥 + 𝑓 5 4 3 2 have given that 𝑎 , 𝑏 , 𝑐 , 𝑑 , 𝑒 , and 𝑓 are all real?

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

Q8:

Are the roots of the equation 3 𝑥 + 2 4 𝑥 + 4 8 = 0 2 real and different?

  • Ano
  • Byes

Q9:

Given that 𝑓 ( 𝑥 ) = 𝑎 𝑥 + 𝑏 𝑥 + 𝑐 2 has a zero at 3 − 4 𝑖 and 𝑓 ( 0 ) = 1 0 0 , determine the values of 𝑎 , 𝑏 , and 𝑐 .

  • A 𝑎 = 3 , 𝑏 = − 4 , 𝑐 = 1 0 0
  • B 𝑎 = − 1 4 . 2 9 , 𝑏 = 8 5 . 7 4 , 𝑐 = 1 0 0
  • C 𝑎 = 4 , 𝑏 = 2 4 , 𝑐 = 8 0
  • D 𝑎 = 4 , 𝑏 = − 2 4 , 𝑐 = 1 0 0
  • E 𝑎 = − 1 4 . 2 9 , 𝑏 = − 8 5 . 7 4 , 𝑐 = − 1 0 0

Q10:

If 7 and 6 are the roots of the equation 𝑥 + 𝑎 𝑥 + 𝑏 = 0 2 , what are the values of 𝑎 and 𝑏 ?

  • A 𝑎 = 4 2 , 𝑏 = − 1 3
  • B 𝑎 = 1 3 , 𝑏 = 4 2
  • C 𝑎 = 2 3 , 𝑏 = 4 2
  • D 𝑎 = − 1 3 , 𝑏 = 4 2
  • E 𝑎 = − 2 3 , 𝑏 = − 4 2

Q11:

If the roots of the equation 𝑥 + 1 3 𝑥 + 𝑚 = 0 2 have a difference of 3, what is the value of 𝑚 ?

Q12:

Find the quadratic equation whose roots are 9 + 7 𝜔 𝑖 and 9 + 7 𝜔 𝑖 2 .

  • A 𝑥 − ( 1 8 − 7 𝑖 ) 𝑥 + 3 2 + 6 3 𝑖 = 0 2
  • B 𝑥 + ( 1 8 − 7 𝑖 ) 𝑥 + 3 2 − 6 3 𝑖 = 0 2
  • C 𝑥 + ( 1 8 − 7 𝑖 ) 𝑥 + 3 2 + 6 3 𝑖 = 0 2
  • D 𝑥 − ( 1 8 − 7 𝑖 ) 𝑥 + 3 2 − 6 3 𝑖 = 0 2

Q13:

How many real solutions does the equation 4 𝑥 + 4 𝑥 = − 1 2 have?

Q14:

Determine the type of the roots of the equation ( 𝑥 − 𝑚 ) ( 𝑥 − 𝑛 ) − 4 6 = 0 , if 𝑚 and 𝑛 are real numbers.

  • Areal
  • Bcomplex and not real

Q15:

Let 𝜔 be a complex cube root of unity. Form a quadratic equation whose roots are 1 − ( 1 + 𝜔 ) − 1 − 1 and 1 − 1 + 𝜔 2 − 1 − 1 .

  • A 𝑥 − 𝑥 − 1 = 0 2
  • B 𝑥 + 𝑥 + 1 = 0 2
  • C 𝑥 + 𝑥 − 1 = 0 2
  • D 𝑥 − 𝑥 + 1 = 0 2
  • E 𝑥 + 2 𝑥 − 1 = 0 2

Q16:

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

Q17:

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

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

Q18:

Determine the quadratic equation whose roots are 2 + 2 𝜔 + 𝜔 2 3 and − 4 + 5 𝜔 − 4 𝜔 2 3 .

  • A 𝑥 − 2 8 𝑥 + 2 7 = 0 2
  • B 𝑥 + 7 2 8 𝑥 − 7 2 9 = 0 2
  • C 𝑥 + 2 8 𝑥 + 2 7 = 0 2
  • D 𝑥 − 7 2 8 𝑥 − 7 2 9 = 0 2

Q19:

Find the quadratic equation whose two roots are − 4 1 + 𝜔 and − 4 1 + 𝜔 2 .

  • A 𝑥 + 1 6 = 0 2
  • B 𝑥 − 4 𝑥 + 1 6 = 0 2
  • C 𝑥 + 8 𝑥 + 1 6 = 0 2
  • D 𝑥 + 4 𝑥 + 1 6 = 0 2

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