TITLE: Can all real polynomials be factored into quadratic and linear factors?
QUESTION [42 upvotes]: So I understand how to do integration on rational functions with a linear and a quadratic denominator, and I understand how to do a partial fraction decomposition, but I was wondering what happens if the polynomial is higher degree and can't be factored. Later in the page, it says this:

However, it can be shown that any polynomial with real coefficients is a product of linear and/or irreducible quadratic factors with real coefficients.

And I was wondering, how do we know this and is this definitely true?

REPLY [2 votes]: This is what  Carl Friedrich Gauss  proved in his 25-page doctoral dissertation at the age of 22. The title of his dissertation is 'New Proof of the Theorem That Every Algebraic Rational Integral Function In One Variable can be Resolved into
Real Factors of the First or the Second Degree." In the foreword to the translation, it is mentioned that this was the first proof of the statement as all other earlier proofs were flawed. 
See http://archive.larouchepac.com/files/pdf/gauss_fundamental_1.pdf