Fundamental theorem of algebra

[ehrenfest]

Homework Statement

There are two versions of the fundamental theorem of algebra, one that says a polynomial of degree n has n roots and the other that says a polynomial can be factored into linear and irreducible quadratic factors. Is there a quick way to see how they are the same?

Homework Equations

The Attempt at a Solution

 

[Gib Z]

The 2nd version is talking within the real numbers ie A polynomial with real coefficients can be factored into linear and irreducible quadratic factors over R.

The first version is the same because the factor theorem says that for P(x), if g is a root then (x-a) is a factor. We can use the quadratic formula too see that any quadratic factor can be factored into linear factors, if factored over C instead of R.

 

[ehrenfest]

I see why the second version implies the first version. I do not see why the first version implies the second version.

How do you know that you can get rid of all of the factors (x - a) where a is complex, since the 2nd version really says that A polynomial with real coefficients can be factored into linear and irreducible quadratic with real coefficients.

 

[Gib Z]

O yes I forgot about that implication. Remember the complex conjugate theorem, which states the for polynomials with real coefficients, complex roots will come in conjugate pairs.