Using the Location Principle to Identify Zeros of Polynomial Functions

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will look at polynomial functions and their zeros. We will look at real world and mathematical examples to define and discuss how to use the location principle to approximate zeros of polynomial functions.

The Location Principle

Suppose that there is a forest near your house that has recently become populated with a rare breed of squirrels. Scientists have tracked the population of the squirrels over the past year, and by putting natural resources and space into consideration, they've come up with the following function to model how the squirrel population will change over time.

  • P(x) = - x2 + 47x + 12, where x = number of months since the scientists started observing the squirrels.

Notice that P(x) is a polynomial function. A polynomial is a mathematical expression of a sum of terms containing the same variable raised to different powers.

These squirrels are really cute, so you're hoping they'll be around for a while. This makes you wonder when the squirrels will be gone from the forest based on the population model. When will the population reach zero? Mathematically speaking you want to know the zeros of the polynomial.

The zeros of a polynomial are the x-values that give a y value of zero when plugged into the polynomial. Graphically, the zeros of a polynomial function are where the function crosses the x-axis because this is where the function value is equal to zero.

Knowing this, you draw up a graph of the scientists' function, and look to see where it crosses the x-axis.


It looks like the squirrel population will hit zero sometime between 45 and 50 months, but it's hard to pinpoint the exact number of months.

Notice that when the function's graph crosses the x-axis at the zeros of the function, the y value either changes from positive to negative or from negative to positive. This leads to a useful rule called the location principle.

The location principle for zeros of polynomial functions states that if we have a polynomial function f(x) and if f(a) > 0 and f(b) < 0, then f(x) has at least one zero between a and b. That's fancy talk for the fact that if a polynomial function's value changes signs between x = a and x = b, then the function must have a zero between x = a and x = b.

To illustrate this, consider that it looks like the squirrel population will hit zero sometime between 45 and 50 months. If we plug these values into the population function, we get that P(45) = 102 and P(50) = -138.


Notice that the function's value changes from positive to negative between x = 45 and x = 50. Therefore, the polynomial function has a zero between x = 45 and x = 50.

Approximating Zeros From a Table

Now, as we said, it's hard to pinpoint the exact number of months based on the graph, and a 5-month span from 45 months to 50 months is quite a bit of time. Thankfully, we can use the location principle to narrow it down further and approximate more accurately where the zero of the polynomial function will occur.

We know that a zero will occur between any x-values a and b, where the function value changes sign, so we can make a table of function values for x-values 45 through 50, and then look for a sign change between function values. Then we will know that a zero occurs between the x-values that correspond to the function values where the sign change took place.

x P(x)
45 102
46 58
47 12
48 -36
49 -86
50 -138

From the table, we see that P(47) = 12 and P(48) = -36, and the function value changes from positive to negative between x = 47 and x = 48. Therefore, by the location principle, we know the polynomial function has a zero between x = 47 and x = 48. This tells us the squirrels will be completely gone in 47-48 months. That certainly narrows it down more accurately!

Another Example

Let's take a look at one more example. Suppose that we are working with the following polynomial function:

f(x) = x3 + 5x2 - 3x - 10

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