We never discussed how to calculate \(||f-p||_{\infty}\).
Maple has a function to do this, but it can be inaccurate. Most people will use Sollya for this. It will give an interval guaranteed to contain the supremum norm. A sketch of the method is provided in the book. The basic idea is to use the triangle inequality for \(||(f-T)+(T-p)||_{\infty}\), where \(T\) is usually something like a Taylor approximation for which a bound on \(\supf{f}{T}\) is known theoretically.