Factoring Polynomials with Real Coefficients

[phyguy321]

Homework Statement

Prove that every polynomial with real coefficients factors into linear and quadratic real factors.

Homework Equations

So show that f(x) = x^{n} + a_{}n-1x^{n-1}+...+ax +a_{}0 factors into some (x-r_{}1)(x^{2}+4)...

The Attempt at a Solution

I understand how and why this is true but could someone point me in the right direction on where to start?

 

[Dick]

If all of the roots r_i are real then you are done. Suppose there is a factor (x-a) with 'a' complex? You can match that up with another factor so that the product is a real quadratic. What's that other factor and how do you know it exists?