Roots of Polynomials and Roots of Unity · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Every polynomial meets its roots

Mathematics · Algebra · ages 17-18
Name ______________________   Date ____________

  1. A ninth root of unity is plotted at Argand coordinates (a, b). Which relation must hold?

    • a² + b² = 9
    • a² + b² = 1
    • a² + b² = 81
    • a² + b² = 0

  2. How many distinct complex roots does (x - 3)²(x² + 4) = 0 have?

    Answer: ______________

  3. x cubed minus 2x squared plus x minus 2 factors as (x minus 2)(x squared plus 1). What are its three roots?

    • 2, i and -i
    • 2, 1 and -1
    • 2 only

  4. Which of these is a 4th root of unity?

    • 1 + i
    • 2
    • i

  5. A degree 7 polynomial has exactly three distinct complex roots. Two have multiplicities 2 and 3. What is the multiplicity of the third root?

    Answer: ______________

  6. Which statement is guaranteed for a degree 6 polynomial with complex coefficients?

    • It has exactly six different real roots in every case.
    • It has exactly six different non-real roots in every case.
    • Its degree alone determines the value of every root.
    • It has six complex roots counting multiplicity.

  7. For f(x) = x³ + 4x² + 36x + 144, f(-4) = 0. Divide out the corresponding linear factor to find all roots.

    • -4, 6i and -6i
    • -4, 36i and -36i
    • 4, 6i and -6i
    • -4, 6 and -6

  8. Solve z⁴ = 1 using exponential form. Which row lists all arguments in 0 ≤ θ < 2π and their corresponding Argand coordinates?

    • Arguments: 0, π/2, π, 3π/2; points: (1, 0), (0, 1), (-1, 0), (0, -1).
    • Arguments: 0, π/2, π, 3π/2; points: (2, 0), (0, 2), (-2, 0), (0, -2).
    • Arguments: 0, π, 2π, 3π; points: (1, 0), (-1, 0), (1, 0), (-1, 0).

  9. Why must a real cubic always have a real root?

    • Every root of a real cubic must itself be a real number
    • Its continuous graph has ends in opposite directions
    • Its leading coefficient must equal 1 for every real cubic

  10. A cubic with real coefficients has roots -1 and 3 + 4i. What is its third root?

    • -3 - 4i
    • -3 + 4i
    • 3 - 4i
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Answer key

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Every polynomial meets its roots W1-mt_XAISygqdQr-s1

  1. a² + b² = 1 · Every root of unity lies on the unit circle. Its distance from the origin is 1, so its coordinates satisfy a² + b² = 1.
  2. 3 · The root 3 repeats, while x² + 4 gives 2i and -2i. There are three distinct roots and four roots counting multiplicity.
  3. 2, i and -i · The linear factor gives 2 and the quadratic gives plus and minus i.
  4. i · A fourth root of unity has fourth power 1. Here i⁴ = 1, (1 + i)⁴ = -4 and 2⁴ = 16.
  5. 2 · The multiplicities total 7. The known roots account for 2 + 3 = 5, leaving multiplicity 7 - 5 = 2 for the third root.
  6. It has six complex roots counting multiplicity. · The fundamental theorem gives six complex roots counting repetitions. It does not promise distinct roots, real roots, non-real roots or their values.
  7. -4, 6i and -6i · The root -4 gives factor x + 4. Division leaves x² + 36, so the other roots are 6i and -6i.
  8. Arguments: 0, π/2, π, 3π/2; points: (1, 0), (0, 1), (-1, 0), (0, -1). · The modulus is 1 and the arguments are 2πk/4 for k = 0, 1, 2, 3. The roots are 1, i, -1 and -i, at the four listed unit-circle coordinates.
  9. Its continuous graph has ends in opposite directions · A real cubic is continuous and has outputs of opposite signs far to the left and right. Between those outputs it must take the value zero.
  10. 3 - 4i · Real coefficients require the conjugate of 3 + 4i as another root. Keep the real part 3 and negate the imaginary part to obtain 3 - 4i.
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