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I want to restrict myself to polynomials with real-valued coefficients and consider only real-valued roots (so, in this context, for instance, a 2nd degree polynomials sometimes cannot be factorized).

In this context, can a 4th degree polynomial always be factorized in two 2nd-degree polynomials ?

Or is it possible to come up with a 4th degree polynomials which could not be factorized into any real-coefficients polynomials ?

(and - this maybe should be the object of a further question - but could we extend this property to all even degree polynomials ?)

xdutoit
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  • The answer is yes, since real polynomials of degree higher than $2$ are always reducible. So you can factor into factors of degree two by multiplying pairs of degree 2 factors together. – Rushabh Mehta Dec 19 '19 at 15:24
  • Every polynomial with real coefficients can be factored into linear and quadratic polynomials with real coefficients. This has essentially been known for several hundred years. However, the coefficients of the factors are often not expressible in "nice ways" in terms of the original polynomial's coefficients. For example, even if the original polynomial's coefficients are integers, the factored coefficients might not be expressible in terms of radicals. – Dave L. Renfro Dec 19 '19 at 15:29
  • Regarding methods of factoring a 4th degree polynomial as two quadratic polynomials, see 3. Quadratic Factors of $;x^4 + 10x^2 - ; 96x - 71$ in this answer. – Dave L. Renfro Dec 19 '19 at 15:32

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The ring of one-variable polynomials over any field is a Euclidean ring (it means that the divisibility properties in it are somewhat similar to the ones in $\mathbb{Z}$). Particularly, that means, that all polynomials can be factorized in a product of "prime" polynomials (usually called "irreducible"), that can not be divided by any polynomial other then a non-zero constant or themselves multiplied by a non-zero constant (non-zero constant polynomials play here a role similar to the one of $\{1; -1\}$ in $\mathbb{Z}$). The irreducible polynomials over $\mathbb{R}$ are well classifies. There are two classes of them:

1)$ax + b$

2)$ax^2 + bx + c$, where $b^2 < 4ac$

Thus there are four options for our fourth degree polynomial. It is either $(ax^2 + bx + c)(dx^2 + ex + f)$ (with $b^2 < 4ac$ and $e^2 < 4df$) or $(ax^2 + bx + c)(dx + e)(fx + g)$ (with $b^2 < 4ac$) or $(ax+b)(cx+d)(ex + f)(gx + h)$.

Chain Markov
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