I have written a program which finds the roots of polynomial using Newton's Method. After finding the first root to within a tolerance (note that this also finds complex roots), I use synthetic division to remove that root from the original polynomial (f = f/(x-root))

My question is, how does this affect the error? I can tell I get some shift as I look at my 20th root, but exactly how would I quantify this, and how would I ensure that the max error is still less than my tolerance?