Can anyone give an intuitive proof of the theorem on polynomial factorization?

[Shahed al mamun]

Recently I have known an algebraic theorem on polynomial while learning the method of partial fraction.
The theorem is :
Any polynomial can be written as the product of linear factors and irreducible quadratic factors.
I did not find an intuitive proof of this theorem.I asked a question in this link http://math.stackexchange.com/questions/1158560/need-of-an-intuitive-proof-of-an-algebraic theorem

and an answer was given there but I did not fully understand this.Can anyone prove the theorem intuitively?

 

[mathman]

The theorem you stated applies to polynomials with real coefficients and the factoring terms all involve real numbers. The fundamental theorem of algebra states that the polynomial can be expressed as a product of linear terms involving the roots (times the lead coefficient). Since the coefficients are all real, complex roots will appears as conjugate pairs. Combining a pair gives an irreducible quadratic term with real constants.