Zero Analysis: Definition & Concept

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Zero analysis is a way of figuring out the answers to certain functions. In this lesson, you will see how to find the zeroes of polynomial functions. Learn how synthetic division helps you out.


Zero analysis means that you are finding all the zeroes or solutions to a particular function. The usual functions that you will work with are polynomials of the following form.

A polynomial function
zero analysis

The functions you will be solving will look like the following.

Example polynomials
zero analysis

These polynomials are not the easiest to solve for, but I'm going to show you the method to use to find the zeroes or solutions of any polynomial. The steps I will show you apply to solving all polynomial functions. Once you master it, it will serve you well. So put on your thinking cap and we can get to work.

Setting up the Problem

What you will see might seem tedious, but you will see how it will shorten your problem later.

We will go ahead and solve the first function I showed you.

Our problem polynomial function
zero analysis

Our first step is to look at the coefficients of the first and last terms. Coefficients are the numbers you see in front of the variables in each term. Terms are the product of a number coefficient and variables and are separated from one another by a plus or minus sign. In this case, we have a 1 as our coefficient for the x^3 term and our last term has a coefficient of -8.

In setting up our problem, we want to find all the factors of our two coefficients. We will form fractions from these factors. Factors are the numbers used to multiply to get to our desired number. They are also the numbers that divide evenly into our number. So factors of -8 will be the numbers we use to multiply to get to -8 or numbers that divide evenly into -8. Our factors of -8 will go in the numerator and our factors of 1 will go in the denominator. For all of our factors, we will include both the positive and negative because if we switch the signs, we will still be able to multiply to -8. Let's see what we get.

Finding the fractions
zero analysis

For the factors of -8, I have the negative and positive of 1, 2, 4, and 8. Notice how when you multiply 2 and -4 you get -8, and also if you multiply -2 and 4, you also get -8. This is why we include both positive and negative versions, because the signs can easily be switched and we still can multiply to our desired number. We also have placed our factors of -8 in the numerator of our fractions and the factors of 1 in the denominator. We have simplified the fractions as well. We know that anything over 1 is itself, so our fractions simplified to the whole numbers.

The purpose of these fractions is to give us a list of possible answers. The way polynomials work is that when you get this list of fractions, all the solutions will come from this list. Not all the fractions will be a solution, but some of them will be.

What we do with this list of fractions is to choose one number at a time from the list to see if it is a solution of our polynomial. We check the fraction by plugging it into the function to see if the function equals zero. Let's choose 1 as our first number to try. Let's see if it zeroes out the function.

Is 1 a zero of our function?
zero analysis

Look at that! 1 is a solution of our function. That means a factor of our function is (x-1). When we write our solution in parentheses form with our variable, we use a minus sign if our solution is positive and a plus if our solution is negative. Our solution is positive, so we use a minus sign. Because we know that (x-1) is a factor of our function, we can divide our function by that factor to start simplifying our function. We will use synthetic division.

Synthetic Division

Synthetic division is another method of dividing polynomials but only works when you are dividing by factors such as (x-a) where a is a number. What we do with synthetic division is we put our zero on the left side and we draw division brackets. Next, we place our coefficients in order under the bracket. Our zero is 1 and our coefficients for our function are 1, -7, 14, and -8. If we were missing one of our terms - for example, we didn't have an x^2 term - we would place a zero in its place. For our function, because all of our terms are present, we don't need any additional zeroes.

Synthetic division set up
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The way synthetic division works is you draw another line underneath leaving space to write numbers between the line and under our coefficients. We bring down our first coefficient and we multiply it by the zero and we place that number under the next coefficient.

Synthetic division in progress part 1
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Next, we add those two numbers and write it under the line. Then we take the sum and we multiply it by the zero and we place it under the next coefficient.

Synthetic division in progress part 2
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We continue this process until we reach the end.

Synthetic division in progress part 3
zero analysis

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