By the Rational Root Theorem, the possible rational roots are
±(1, 2, 3, 4, 6, 12, 1/5, 1/7, 1/35, 2/5, 2/7, 2/35, 3/5, 3/7, 3/35, 4/5, 4/7, 4/35, 6/5, 6/7, 6/35, 12/5, 12/7, 12/35). If we let
f(x) = the given polynomial, we find that f(1) is positive and f(-1) is negative, so there is at least one real root between 1 and -1. Testing the rational possibilities that lie in this interval, 2/5 is a root.
So, x - 2/5 is a factor of f(x). Dividing out by x - 2/5, we get
f(x) = (x - 2/5)(35x2 - 30) = (5x - 2)(7x2 - 6)
= (5x - 2)(√7x - √6)(√7x + √6)
