IGNOU MTE 6 2026 - Show that there are infinitely many values of for which&nbs

Question

Show that there are infinitely many values of $\alpha$ for which $x^7 + 15x^2 - 30x + \alpha$ is irreducible in $\mathbb{Q}[x]$ .

09 Jan 2026

Answer :

Word Count : 133

Consider the polynomial (f(x)=x^{7}+15x^{2}-30x+\alpha\in \mathbb{Q}[x]). Apply Eisenstein’s criterion with the prime (p=5). The leading coefficient (1) is not divisible by (5). The coefficients of (x^{2}) and (x) are (15) and (-30), both divisible by (5). If (\alpha) is chosen to be divisible by (5) but not by

_________ __________ ________ _________ _________ ___ _________ ______ ___ ___ _______ _________.
_______ ___ ______ ___ _________ ____ ____ ___ __________ ___.
_____ _________ _____ ________ ________ __________ _______ ___ __________ ________ _____ _____.
_________ _____ ________ ______ ________ _________ _______ ____ ____ __________ _______ ____.
______ ______ _________ _____ ___ _______ __________ ___ _________ _______ _________ ________.
_______ _______ ________ _______ ______ _____ ____.
____ ________ ________ ___ _________ ____ _______ ______ ______ ___ ____ __________.
__________ _________ _________ ________ _____ ______.
_______ ________ __________.

IGNOU ACADEMY

Your IGNOU journey, made easier.