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Geometry of Complex Roots

Roots equally spaced around circles.

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Geometry of Complex Roots

To graph the roots of a polynomial, first you must find one root. You only need one! Plot this root on the graph, draw a circle around the origin touching the root (now a point on your circle), and figure out how many degrees apart each root is using this formula:   \begin{align*}\frac{2\pi}{n}\end{align*} . 

The roots will be evenly spaced along the edge of this circle!

Remember: The number of roots is the power of the polynomial!


  1. What is the spacing in polar coordinates between the roots of the polynomial \begin{align*}x^6 = 12\end{align*} ?
  2. How many roots does \begin{align*}x^{10}=1\end{align*} have?
  3. Calculate the roots of \begin{align*}x^{10}=1\end{align*} and represent them graphically.
  4. How many roots does \begin{align*}x^4=16\end{align*} have?
  5. Calculate the roots of \begin{align*}x^4=16\end{align*} and represent them graphically.
  6. Describe how to represent the roots of \begin{align*}x^6=1\end{align*} graphically without first solving the equation.

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