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

We can use the following result, called the Location Principle, to help us find zeros of polynomial functions:[br][br]If [math]f[/math] is a polynomial function and [math]a[/math] and [math]b[/math] are two numbers such that [math]f\left(a\right)<0[/math] and [math]f\left(b\right)>0[/math], then [math]f[/math] has at least one real zero between [math]a[/math] and [math]b[/math] .

Find all real zeros of the polynomial function[br][math]f\left(x\right)=6x^3+5x^2-17x-6[/math][br][br]The following steps show how to use the Location Principle to find the real zeros of polynomial functions.[br][br]Step 1: Enter values for [math]x[/math].[br]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 [math]x[/math] fills in from 0 to 5.[br][br]Step 2: Enter values for [math]f\left(x\right)[/math].[br]In cell B2 under [math]f\left(x\right)[/math], 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 [math]f\left(5\right)[/math].[br][br]Step 3: Use the Location Principle:[br]The spreadsheet in step 2 shows that [math]f\left(1\right)<0[/math] and [math]f\left(2\right)>0[/math]. So, by the Location Principle, [math]f\left(x\right)[/math] has a zero between 1 and 2. [br]Now that we have an idea where a zero is located, we have a few options available for us to locate the zero:[br]1. We could remake our spreadsheet to go in steps of 1/2 instead of 1.[br]2. We could graph the function and read the zero from the graph.[br]3. We can use the rational zero theorem. If we choose this option, we find that a zero occurs at [math]x=\frac{3}{2}[/math]. Synthetic division confirms this and enables us to factor the polynomial completely. Do this in the exercise that follows.