How to Find the Zeros of a Function

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When working with functions, you sometimes need to calculate the points at which the function's graph crosses the x-axis. These points occur when the value of x is equal to zero and are the zeroes of the function. Depending on the type of function you're working with and how it's structured, it may not have any zeroes, or it may have multiple zeroes. Regardless of how many zeroes the function has, you can calculate all of the zeroes in the same way.

TL;DR (Too Long; Didn't Read)

Calculate the zeroes of a function by setting the function equal to zero, and then solving it. Polynomials may have multiple solutions to account for the positive and negative outcomes of even exponential functions.

Zeroes of a Function

The zeroes of a function are the values of x at which the total equation is equal to zero, so calculating them is as easy as setting the function equal to zero and solving for x. To see a basic example of this, consider the function f(x) = x + 1. If you set the function equal to zero, then it will look like 0 = x + 1, which gives you x = -1 once you subtract 1 from both sides. This means that the zero of the function is -1, since f(x) = (-1) + 1 gives you a result of f(x) = 0.

While not all functions are as easy to calculate zeroes for, the same method is used even for more complex functions.

Zeroes of a Polynomial Function

Polynomial functions potentially make things more complicated. The problem with polynomials is that functions containing variables raised to an even power potentially have multiple zeroes since both positive and negative numbers give positive results when multiplied by themselves an even number of times. This means that you have to calculate zeroes for both positive and negative possibilities, though you still solve by setting the function equal to zero.

An example will make this easier to understand. Consider the following function: f(x) = x2 - 4. To find the zeroes of this function, you start the same way and set the function equal to zero. This gives you 0 = x2 - 4. Add 4 to both sides to isolate the variable, which gives you 4 = x2 (or x2 = 4 if you prefer to write in standard form). From there we take the square root of both sides, resulting in x = √4.

The issue here is that both 2 and -2 give you 4 when squared. If you only list one of them as a zero of the function, you're ignoring a legitimate answer. This means that you have to list both of the zeroes of the function. In this case, they are x = 2 and x = -2. Not all polynomial functions have zeroes that match up so neatly, however; more complex polynomial functions can give significantly different answers.

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