# Showing irreducibility of $f(x)=a_nx^n+a_{n-1} x^{n-1} +.....+a_1x+a_0$ over $\mathbb{Z}[X]$ when $|a_0| > |a_1|+|a_2|+|a_3|+.....+|a_n|$ [duplicate]

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.

• 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. Apr 9 at 5:10
• Thank you @Sarvesh Ravichandran Iyer. Yes it helped me. Apr 9 at 14:44