Let $f(x)=a_nx^n+a_{n-1} x^{n-1} +.....+a_1x+a_0$ be a polynomial with integer coefficients. If $a_0$ is prime and $|a_0| > |a_1|+|a_2|+|a_3|+.....+|a_n|$. Show f(x) is irreducible over $\mathbb{Z}[X]$

I observed that we could not apply Eisenstien's Theorem here. So I decided to follow the traditional method of contradiction. I assumed $f(x) = g(x)h(x)$

Then I noticed that since $a_0$ is prime. So in one of the polynomials $g(x), h(x)$ the ending coefficient is $1$ and the other is $a_0$

But then I am not able to proceed further. Can someone please help me.

  • 3
    $\begingroup$ Does Show that $f(x)$ is irreducible answer your question? The question combines with the small clarification provided in an answer to solve your question. This technique is referred to as root location. $\endgroup$ Apr 9 at 5:10
  • $\begingroup$ Thank you @Sarvesh Ravichandran Iyer. Yes it helped me. $\endgroup$
    – Gloona
    Apr 9 at 14:44


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