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Worksheet: Real and Complex Roots of Polynomials

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