Please help me with my abstract algebra math question Please

Please help me with my abstract algebra math question. Please provide all steps.

polynomials in the given rings klzl 4. (6 points) Determine if the following are irreducible

Solution

f(x) = x⁶-210x -616   

using Descart\'s rule of sign we can find out how many real roots are there   

sign of f(x) : + - -

there is exactly one change in the sign and there is exactly one + real root for this eqn

the root lies between 3 .3 and 3.35 as f(3.3) -18<0 and f(3.35) =94 >0

the root is real . it may be rational or irrational

hence factorisation is not possible in Q

there is 1 -ve real root as sign of f(-x) : + + - there is one change in sign

the given eqn has 1 +ve , 1 -ve real root and the remaining 4 roots are imaginary