What are the roots of a function?

A function has a root when it crosses the x-axis, i.e. . A function can have more than one root, when there are multiple values for that satisfy this condition. The goal is to find all roots of the function (all values). In general we take the function definition and set to zero and solve the equation for .

Root of a linear function

Consider a linear function . We want to find the root by setting to zero:

(1)  

We found that this function has a root for , meaning that it crosses the x-axis and the coordinate . The graph illustrates this:

Root of a quadratic function

Consider the quadratic function (polynomial of second degree) . We want to find the root by setting to zero and solving the equation for :

(2)  

We divided the equation by 2 to bring it into the monic form (, where and ), so that it can be easily solved using the quadratic formula .

(3)  

We found that this function has two roots, at and at . The following graph illustrates this:

Roots of a polynomial of third degree

Consider the function . Again we set to zero and solve the equation for .

(4)  

In the last step we factored out . This means that this function is zero, when or when . We already solved the second part in the example of a quadratic equation. Thus, this function has three roots at , and . The following graph illustrates this: