Factoring polynomials into linear quadratic factors | Linear Algebra MATH1141 | N J Wildberger

Following on from the (necessarily approximate) Fundamental theorem of Algebra, we investigate real polynomials and their (necessarily approximate) factorization into linear and quadratic factors. To clarify this we look more carefully at cyclotomic factorizations of z^n-1 and the connections with roots of unity.

We also explain the computational hurdles in applying these theoretical results to actual real polynomials.

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[raghebalghezi]

Thank you very Professor! Very useful content.

[devrimturker]

There is a concept "algebraically indistinguishable" on complex conjugates.

[postbodzapism]

I was thinking, like can we be serious on the words "approximating" or "we can approximate".
Like the FTA can be interpreted as
Given a non-constant polynomial p(z) of degree d and an extent of accuracy n, we can always find d complex numbers z = a+bi (with multiplicity counted) such that if the decimal expansion of a is the number
x1.x2x3x4...xn × 10^m, then p(x1.x2...x{n-1}y + bi) is closest to zero when y is chosen to be xn among the 10 possible choices 0, 1, 2, ..., 9 of y, for a fixed b, and vice versa for b.

[Kindiakan]

Hi Prof, does what you teach in this video conflict your convictions on the existence of real numbers? If so, how do you reconcile the gap?

[carlyet]

:(, I can't get the video component - please use an open codec - I can just follow on the audio only. thank you.