x7 3x3 1 0 has no rational solutions Problem from An Intr

x^7 + 3x^3 + 1 = 0 has no rational solutions.

Problem from (An Introduction to Number Theory with Cryptography by Krafts and Washignton.) Section 3.3, problem 10.

Solution

According to Rational root theorem, a polynomial equation
a_nxⁿ + a_n-1x^n-1 + ... + a₀ = 0 will have rational roots in the form of x = p/q where p is divisor of a₀ and q is integer factor of a_n
In given polynomial x⁷ + 3x³ + 1 = 0 , p/q can be {-1,+1}
Both {-1,+1} doesn\'t satisfy our polynomial. Hence No Rational solutions.