Polynomial Shapes

Degree, roots and turns

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

• A degree-n polynomial has at most n real roots and at most n − 1 turns.
• The leading term decides what happens at the ends.
• Remainder theorem: dividing $p(x)$ by $(x - a)$ leaves $p(a)$.

Steps

1. Roots and wiggles
2. How many roots?
3. End behaviour
4. The factor theorem
5. Turning points
6. Remainder theorem