Since this problem specifically asks about rational roots, you are clearly expected to know, and use, the "rational root theorem", that skeeter referred to. It is this:

"if the polynomial equation \(\displaystyle a_nx^n+ a_{n-1}x^{n-1}+ \cdot\cdot\cdot+ a_1x+ a_0= 0\), with integer coefficients, has the rartional number \(\displaystyle \frac{p}{q}\) as a solution, then the denominator, q, must evenly divide the leading coefficient, \(\displaystyle a_n\), and the numerator, p, must evenly divide the constant term, \(\displaystyle a_0\).

In this problem the leading coefficient is 1 so the denominator of any rational root must evenly divide 1 which means it must itself be 1 or -1 which means any rational root must be an integer. The constant term is 3 which has, as factors, only 1, -1, 3, or -3.

Zaqiqu, you understand that "Newton's method" only gives a numerical **approximation**, not an exact solution, and so is not appropriate for this problem, don't you?