One of the most fundamental problems in scientific
computing is solving equations of the form *f*
(*x*) = 0. This is often referred to as finding the
*roots*, or
*zeros*, of *f*
(*x*). Here, we are interested in the real roots of *f
* (*x*), as opposed to any complex roots
it might have. Specifically, we will focus on finding real roots when
*f* (*x*) is a
polynomial.

Although factoring and applying formulas are simple
ways to determine the roots of polynomial equations, a great
majority of the time polynomials are of a large enough degree and
sufficiently complicated that we must turn to some method of
approximation. One of the best approaches is *Newton's
method*. Fundamentally, Newton's method looks for a root
of *f * (*x*) by moving closer
and closer to it through a series of iterations. We begin by
choosing an initial value *x* =
*x* _{0} that we think is
near the root we are interested in. Then, we iterate using the
formula:

until *x _{i} *

The derivative of a function is fundamental to calculus and can be described in many ways. For now, let's simply look at a formulaic description, specifically for polynomials. To compute the derivative of a polynomial, ...

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