The values of various roots can be approximated using Newton's method. For example, to approximate the value of \(\sqrt[3]{10}\), let x = \(\sqrt[3]{10}\) and cube both sides of the equation to obtain \(x^3 = 10\), or \(x^3 - 10 = 0\). Therefore, \(\sqrt[3]{10}\) is a root of \(p(x) = x^3 - 10\), which can be approximated by applying Newton's method. Approximate the following value of r by first finding a polynomial with integer coefficients that has a root r. Use an appropriate value of \(x_0\) and stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding.
\(r = 6\frac{1}{3}\)
