Why Every Polynomial Has a Root

Counting roots by winding

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Fundamental theorem of algebra: a degree-n polynomial has exactly n complex roots.
• The winding number of p(circle) around 0 counts the roots inside the circle.
• Tiny circles wind 0 times, huge ones n times, so n roots must lie in between.

Steps

1. How many roots?
2. Following the image
3. Watch it wind
4. Big circles
5. Why not just the reals?
6. Roots inside a circle