PlaySpectrum

Helper · book companion path

Prior: De Moivre & powers · Next: Turn and stretch (quest) · Shelf: Kurs 4

Math / Helper companion / Polynomials over C

Uncut · book companion · ~239–251

Spend roots Tool — factors & light FTA

Polynomials over C

Score 0 / 100

Spend the powers/roots Tool on polynomials: write linear factors over C, read light FTA (n roots counting multiplicity), meet roots of unity from De Moivre. Deeper algebra leftovers stay Soon. (faktorisering · algebraiska fundamental­satsen · enhetsrötter)

Vortex

V1 — Factor with complex roots.

Builds on: solving zⁿ=a / square roots from powers Tool.

Tool gained: factor quadratics like z²+1 = (z−i)(z+i) over C.

  1. Principle. A polynomial “stuck” over reals can split once roots from the powers Tool are allowed.
  2. Model. If r is a root, (z−r) is a factor (over C).
  3. Prediction / Answer. z²+1=0 → z=±i → (z−i)(z+i). Transfer: resonance poles as factors.
Vortex

V2 — Light FTA: n roots counting multiplicity.

Builds on: factor-via-root Tool (V1) — degree counts factors.

Tool gained: light FTA reading — degree-n polynomial over C has n roots counting multiplicity (statement use, not proof).

  1. Principle. Over C you expect a full set of linear factors — same root Tool, counted to match degree.
  2. Model. p(z)=c(z−r₁)…(z−rₙ) (complex r_k, repeats allowed).
  3. Prediction / Answer. z³−1 = (z−1)(z−ω)(z−ω²) with cube roots of unity from De Moivre. Heavier algebra stays Soon.

1. Poor dump → rewrite

Poor explanation

Book dump: FTA as a slogan with no root Tool. Our rewrite: find roots with De Moivre → write (z−r) → count to degree; cube roots of unity as the picture.

Tool · weapon

Weapons: (z−r) factors from complex roots → light FTA count → roots of unity. Kap 4 core climb complete; heavier poly theory Soon.

2. Hands-on — cube roots of R

Watch the three cube roots of a = R∠0 for varying R (unit angles 0°, 120°, 240°).

…

3. Fake → real

Fake → real

Fake

“Factor z²+1.” No job.

Real

A notch filter needs poles at ±i so the transfer factor includes (z−i)(z+i)=z²+1 — same factor Tool, now a signal job.

Gate — four, all required

1. Reconstruct: z² + 1 factors over C as…

2. Why does factoring build on powers/roots (climb)?

3. Predict: number of roots of z³−1 = 0 in C (counting multiplicity)?

4. Transfer: cube roots of unity sit where?