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March 21st, 2018, 05:57 PM  #1 
Member Joined: Sep 2011 Posts: 97 Thanks: 1  Irreducible or reducible polynomial?
Hi all, I have used rational root test and obtain this result for the question: f(x) = 2x^4+8x^3+5x^2−7x−3=(2x^2+2x−3)(x^2+3x+1) This shows that there is no linear factor but a quadratic factors instead. Do I still consider such polynomial as reducible or irreducible? Am I correct to say that f(x) has no linear factors, hence (2x^2+2x−3) and (x^2+3x+1) have no linear factors. As (2x^2+2x−3) and (x^2+3x+1) are of degree 2, it follows that they are irreducible in Q[x]? 
March 21st, 2018, 07:57 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 350 Thanks: 192 Math Focus: Dynamical systems, analytic function theory, numerics 
It is correct as long as you say it is irreducible over $\mathbb{Q}[x]$ because it has no linear factors in $\mathbb{Q}[x]$. It does have linear factors; every polynomial splits into linear factors over its splitting field. When discussing irreducibility, it is important to be specific about which fields/rings you are referring to. The property of being irreducible is meaningless without specifying a field/ring. It's also worth pointing out, since it's a common mistake, that a polynomial can be reducible over some polynomial ring even if it has no linear factors in that ring. Linear factors are only special when dealing with quadratic polynomials or polynomials of prime degree. Last edited by skipjack; March 24th, 2018 at 03:24 AM. 
March 24th, 2018, 02:52 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,089 Thanks: 846 
'Reducible' does NOT mean "can be factored into linear factors". It simply means "can be factored into a product of polynomials of lower degree than the original polynomial. You are correct that this polynomial cannot be factored (over Q) into linear terms but the fact that it can be factored into the product of two quadratic polynomials means that it is "reducible".

April 14th, 2018, 09:30 AM  #4  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 166 Thanks: 50 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Quote:
 

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