Rational roots theorem?

Question

How does the rational roots theorem show that the roots of this function are all irrational?

f(x)= x^3-30x^2+275x-720

I know rational roots theorem is finding out all possible roots by taking all the factors (p) of the constant, and dividing it by all the factors (q) of the highest power. In otherwords, all possible roots = (p/q)

But since q can only be 1 here, how are all the roots "Irrational"?

Answer 1

You would take all the factors of 720 (because you are dividing all of them by one) and use synthetic division with each factor to show that the remainder is not 0. This shows that none of the roots (all possible factors of 720) are rational.

Answer 2

Possible rational roots are 720, 360,180,90, 45, 9,5,3.

But it can be quickly seen that none of these is a rational root. Therefore all roots must be irrational.

Answer 3

for ironduke: actually you have to verify the negative divisors too