Spreadsheet Activity - Use the Location Principle.

Author:marcien.dentey

Question:

How can we use the Location Principle to Identify the zeros of a polynomial?

Answer:

We can use the following result, called the Location Principle, to help us find zeros of polynomial functions: If

is a polynomial function and

and

are two numbers such that

and

, then

has at least one real zero between

and

Example:

Find all real zeros of the polynomial function

The following steps show how to use the Location Principle to find the real zeros of polynomial functions. Step 1: Enter values for

. In the Spreadsheet portion of the Geogebra Applet below, enter "0" into cell A2. Type "=A2+1" into cell A3. Move the cursor to the bottom right corner of cell A3. When the cursor changes from arrow to a solid +, click-hold your mouse and drag down to cell A7. The values of

fills in from 0 to 5. Step 2: Enter values for

. In cell B2 under

, type "=6*A2^3+5*A2^2-17*A2-6". Repeat the method used in step 1 above to fill in the function values through

. Step 3: Use the Location Principle: The spreadsheet in step 2 shows that

and

. So, by the Location Principle,

has a zero between 1 and 2. Now that we have an idea where a zero is located, we have a few options available for us to locate the zero: 1. We could remake our spreadsheet to go in steps of 1/2 instead of 1. 2. We could graph the function and read the zero from the graph. 3. We can use the rational zero theorem. If we choose this option, we find that a zero occurs at

. Synthetic division confirms this and enables us to factor the polynomial completely. Do this in the exercise that follows.