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Complex Numbers & Root Geometry
Polar form, roots of polynomials, rational functions, and geometry in the complex plane.
- For
- Students meeting complex numbers seriously for the first time, and beyond.
- Focus
- Polar and exponential form · Roots of polynomials · Discriminants · Derivatives of rational functions
Curated prompts
Start here.
Written by RIDA MATH. Each prompt says what a strong contribution looks like.
What the discriminant decides
For , describe in geometric terms what changes in the root positions as the discriminant passes through zero. Plot three examples in the complex plane.
A strong contribution: Real roots merge on the real axis and split into a conjugate pair; the plots are labelled.
Where do the zeros of a derivative sit?
For , find the real zeros of . Express them as and explain the pattern.
A strong contribution: Zeros derived and connected to equally spaced angles around the root at .
Symmetry of the roots of unity
Show that the roots of sum to zero for . Give one algebraic and one geometric argument.
A strong contribution: Uses Vieta's formulas and rotational symmetry of the regular polygon.
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