Newton's Method finds Zeros of Functions

Newton's method helps you find “zeros” of a function – the points where the function's value becomes zero. For example, the polynomial

Newton's method gives yet another way to find the zeros of a function – one that is especially useful for computers and calculators.

Why Find Zeros?

In algebra class we spent a lot of time finding zeros of lines and polynomials and other functions. We sometimes call these zeros the “roots” of the function, implying that they somehow ground and characterize the function. Why do we care so much about finding zeros? What's special about the points where the function crosses the axis? Go ahead and think about that for a moment, then scroll down to see what we thought.

• Solve for a Value
  You can make any “solve for a value” problem (f(x)=a) into a “find a zero” problem – just find the zeros of f(x)-a=0 instead.

• Solve for an Equation
  This is still true even if the “solve-for” value is another equation or system of equations: you can still easily turn it into a zero-finding problem.

• Factoring Polynomials
  Once you know the zeros of a polynomial, you also know how to factor it.

• Find minima and maxima
  The minima and maxima of a function are found where the derivative has zeros ( f´(x)=0).

• Find singular points
  The function 1/f(x) will blow up (diverge to infinity) where f(x) has zeros.

Since so many other problems may be reduced to the simpler “find-the-zeros” problem, we'd like to have several good ways to do so.