Question #133624
Check whether x^5+9x^4+12x^2+6 is reducible over Q.
1
Expert's answer
2020-09-28T19:26:38-0400

Given polynomial is f(x)=x5+9x4+12x2+6f(x) = x^5+9x^4+12x^2+6 .

Compare the polynomial with f(x)=a5x5+a4x4+a3x3+a2x2+a1x+a0f(x) = a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 , we have

a5=1,a4=9,a3=0,a2=12,a1=0,a0=6a_5 = 1, a_4= 9, a_3 = 0, a_2 = 12, a_1 = 0, a_0 = 6 .


There esist a prime number p=3p= 3 which divides all coefficients a4,a3,a2,a1,a0a_4,a_3,a_2,a_1,a_0 , but not the leading coefficient a5a_5 and, p2p^2 not divide the constant term .

Hence, using the Einstein Criteria, given polynomial is irreducible over QQ .


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