Roots of Different Polynomials

Complex Polynomials and the Fundamental Theorem of Algebra
Integer Polynomials and the Rational Roots Theorem
Real Polynomials and the Conjugate Roots Theorem
Roots of Different Polynomials

Abstract

Use CJRT to get another root
Put root in the form of
Long division
Repeat until irreducible, using factoring to find factors
Start with
- All roots in will be linear (CPN), and then if roots aren't in , expand the ones with

Example

Factor into a product of irreducible polynomials in , and .

If you encounter a problem like this, don't do it, unless they give you some hint. This isn't math 425 or whatever. If you try to solve it, it's horrible.
- Wentang Kuo

Given that is a root

For :
Using the Buy One Get One Free Theorem, is also a root (see Conjugate.

By DAP and long division,

For :

Note that is irrational in since it has no real roots and (from the reducibility theorem)

For :
is irrational in since it has no root, and .

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