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

    • i
    • 1 + i
    • 2
    • -1 + i

  2. What is the real root of x cubed minus 8 = 0?

    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. The fundamental theorem says a degree 7 polynomial has how many complex roots, counting multiplicity?

    Answer: ______________

  6. A degree 5 polynomial has how many complex roots, counting multiplicity?

    • 5
    • 3
    • 4
    • It depends on the leading coefficient

  7. 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, i and -i
    • 2 only

  8. How many 6th roots of unity are there, and where do they sit?

    • 6, at the corners of a regular hexagon
    • 3, at the corners of a triangle
    • 5, at the corners of a pentagon

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

    • Cubics have no complex roots
    • Odd-degree ends head opposite ways, forcing a crossing
    • The leading coefficient is always 1

  10. A real cubic has roots 2 and i. Name the third root.

    • -2
    • 2i
    • -i
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Every polynomial meets its roots W1-mt_XAISygqdQr-s1

  1. i · i to the 4th is 1, while none of the other options powers to 1.
  2. 2 · Only 2 cubed is 8 among real numbers; the other two cube roots are complex.
  3. 2, i and -i · The linear factor gives 2 and the quadratic gives plus and minus i.
  4. i · i to the 4th is 1, while the others power far from 1.
  5. 7 · Root count equals degree, so 7 roots with repeats counted.
  6. 5 · The theorem promises exactly as many complex roots as the degree, counting repeats.
  7. 2, i and -i · The quadratic factor gives plus and minus i, and the linear factor gives 2.
  8. 6, at the corners of a regular hexagon · Degree 6 gives 6 solutions, spaced every 60 degrees on the unit circle.
  9. Odd-degree ends head opposite ways, forcing a crossing · Opposite ends trap the axis between them, so the graph must cross it.
  10. -i · Non-real roots of real polynomials arrive in conjugate pairs, so -i completes the trio.
Worksheet · LightMySky