# How can you find approximations to the zeros of a function?

Nov 6, 2017

Use Newton's method to recursively define sequences whose limits are zeros...

#### Explanation:

If $f \left(x\right)$ is a continuous, differentiable function then we can usually find its zeros using Newton's method:

Given an approximation ${a}_{i}$ to a zero of $f \left(x\right)$, a better one is given by the formula:

${a}_{i + 1} = {a}_{i} - f \frac{{a}_{i}}{f ' \left({a}_{i}\right)}$

We can use this formula to recursively define a sequence:

${a}_{0} , {a}_{1} , {a}_{2} , \ldots$

Then the limit of the sequence is a zero of $f \left(x\right)$.

By choosing different values for the initial term ${a}_{0}$, the resulting sequence will tend to other zeros of $f \left(x\right)$.

This method is both easy to apply and generally quite effective with polynomial functions.

It also works with both real and complex zeros.