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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:

Determine the type of the roots of the equation ( 2 π‘₯ βˆ’ 4 ) + 1 7 = 0 2 .

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

Q2:

Find the quadratic equation whose roots are 9 + 7 πœ” 𝑖 and 9 + 7 πœ” 𝑖 2 .

  • A π‘₯ βˆ’ ( 1 8 βˆ’ 7 𝑖 ) π‘₯ + 3 2 + 6 3 𝑖 = 0 2
  • B π‘₯ + ( 1 8 βˆ’ 7 𝑖 ) π‘₯ + 3 2 βˆ’ 6 3 𝑖 = 0 2
  • C π‘₯ + ( 1 8 βˆ’ 7 𝑖 ) π‘₯ + 3 2 + 6 3 𝑖 = 0 2
  • D π‘₯ βˆ’ ( 1 8 βˆ’ 7 𝑖 ) π‘₯ + 3 2 βˆ’ 6 3 𝑖 = 0 2

Q3:

If π‘Ž + 𝑏 𝑖 is a root of the polynomial 𝑓 ( π‘₯ ) , what is the value of 𝑓 ( π‘Ž + 𝑏 𝑖 ) ?

Q4:

Solve the equation π‘₯ + 1 = 0 3 , π‘₯ ∈ β„‚ .

  • A  1 2 + √ 3 2 𝑖 , βˆ’ 1 2 + √ 3 2 𝑖 , βˆ’ 1 
  • B  βˆ’ 1 2 + √ 3 2 𝑖 , βˆ’ 1 2 βˆ’ √ 3 2 𝑖 , 1 
  • C  1 2 + √ 3 2 𝑖 
  • D  1 2 + √ 3 2 𝑖 , βˆ’ 1 , 1 2 βˆ’ √ 3 2 𝑖 

Q5:

Given that 𝑓 ( π‘₯ ) = π‘Ž π‘₯ + 𝑏 π‘₯ + 𝑐 2 has a zero at 3 βˆ’ 4 𝑖 and 𝑓 ( 0 ) = 1 0 0 , determine the values of π‘Ž , 𝑏 , and 𝑐 .

  • A π‘Ž = 3 , 𝑏 = βˆ’ 4 , 𝑐 = 1 0 0
  • B π‘Ž = βˆ’ 1 4 . 2 9 , 𝑏 = 8 5 . 7 4 , 𝑐 = 1 0 0
  • C π‘Ž = 4 , 𝑏 = 2 4 , 𝑐 = 8 0
  • D π‘Ž = 4 , 𝑏 = βˆ’ 2 4 , 𝑐 = 1 0 0
  • E π‘Ž = βˆ’ 1 4 . 2 9 , 𝑏 = βˆ’ 8 5 . 7 4 , 𝑐 = βˆ’ 1 0 0

Q6:

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

  • A no
  • B yes

Q7:

Given that π‘₯ βˆ’ 1 2 1 𝑦 + ( π‘₯ + 1 1 𝑦 ) 𝑖 = 5 ( 1 + 𝑖 ) 2 2 , where π‘₯ and 𝑦 are real numbers, find the value of π‘₯ and the value of 𝑦 .

  • A π‘₯ = 3 , 𝑦 = 3 1 1
  • B π‘₯ = 3 , 𝑦 = βˆ’ 2 1 1
  • C π‘₯ = 6 , 𝑦 = 6 1 1
  • D π‘₯ = 3 , 𝑦 = 2 1 1
  • E π‘₯ = 6 , 𝑦 = 5 1 1

Q8:

Are the roots of the equation 3 π‘₯ + 2 4 π‘₯ + 4 8 = 0 2 real and different?

  • Ano
  • Byes

Q9:

Find the quadratic equation whose two roots are βˆ’ 4 1 + πœ” and βˆ’ 4 1 + πœ” 2 .

  • A π‘₯ + 1 6 = 0 2
  • B π‘₯ βˆ’ 4 π‘₯ + 1 6 = 0 2
  • C π‘₯ + 8 π‘₯ + 1 6 = 0 2
  • D π‘₯ + 4 π‘₯ + 1 6 = 0 2

Q10:

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

  • A cannot say
  • B no
  • C yes

Q11:

If 7 and 6 are the roots of the equation π‘₯ + π‘Ž π‘₯ + 𝑏 = 0 2 , what are the values of π‘Ž and 𝑏 ?

  • A π‘Ž = 4 2 , 𝑏 = βˆ’ 1 3
  • B π‘Ž = 1 3 , 𝑏 = 4 2
  • C π‘Ž = 2 3 , 𝑏 = 4 2
  • D π‘Ž = βˆ’ 1 3 , 𝑏 = 4 2
  • E π‘Ž = βˆ’ 2 3 , 𝑏 = βˆ’ 4 2

Q12:

If the roots of the equation π‘₯ + 1 3 π‘₯ + π‘š = 0 2 have a difference of 3, what is the value of π‘š ?

Q13:

How many real solutions does the equation 4 π‘₯ + 4 π‘₯ = βˆ’ 1 2 have?

Q14:

Determine the type of the roots of the equation ( π‘₯ βˆ’ π‘š ) ( π‘₯ βˆ’ 𝑛 ) βˆ’ 4 6 = 0 , if π‘š and 𝑛 are real numbers.

  • Areal
  • Bcomplex and not real

Q15:

Let πœ” be a complex cube root of unity. Form a quadratic equation whose roots are ο€Ί 1 βˆ’ ( 1 + πœ” )  βˆ’ 1 βˆ’ 1 and ο€» 1 βˆ’ ο€Ή 1 + πœ”   2 βˆ’ 1 βˆ’ 1 .

  • A π‘₯ βˆ’ π‘₯ βˆ’ 1 = 0 2
  • B π‘₯ + π‘₯ + 1 = 0 2
  • C π‘₯ + π‘₯ βˆ’ 1 = 0 2
  • D π‘₯ βˆ’ π‘₯ + 1 = 0 2
  • E π‘₯ + 2 π‘₯ βˆ’ 1 = 0 2

Q16:

Determine the type of the roots of the equation ( π‘₯ βˆ’ 1 0 ) ( π‘₯ + 1 0 ) = 2 ( π‘₯ + 8 ) ( π‘₯ + 6 ) .

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

Q17:

Determine the type of the roots of the equation π‘₯ + 4 π‘₯ + 1 = 3 .

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

Q18:

Determine the quadratic equation whose roots are ο€Ή 2 + 2 πœ” + πœ”  2 3 and ο€Ή βˆ’ 4 + 5 πœ” βˆ’ 4 πœ”  2 3 .

  • A π‘₯ βˆ’ 2 8 π‘₯ + 2 7 = 0 2
  • B π‘₯ + 7 2 8 π‘₯ βˆ’ 7 2 9 = 0 2
  • C π‘₯ + 2 8 π‘₯ + 2 7 = 0 2
  • D π‘₯ βˆ’ 7 2 8 π‘₯ βˆ’ 7 2 9 = 0 2

Q19:

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

  • A βˆ’ π‘Ž βˆ’ 𝑏 𝑖
  • B βˆ’ π‘Ž + 𝑏 𝑖
  • C 𝑏 βˆ’ π‘Ž 𝑖
  • D π‘Ž βˆ’ 𝑏 𝑖
  • E 𝑏 + π‘Ž 𝑖

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