What are the roots of a function?

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

Root of a linear function

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

(1)   equation

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

linearroot

Root of a quadratic function

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

(2)   equation

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

(3)   equation

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

quadraticroot

Roots of a polynomial of third degree

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

(4)   equation

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

polyroot