How do I find the real zeros of a polynomial function on a graphing calculator?

If you are using a TI-83 or TI-84, then you first graph the function by going into Y= and entering the part after the f(x) or y= after Y1.
Now you just need to hit the “graph” key, and you should see a nice graph laid out there.
Now what you are looking for are the zeroes. These are the points where the graph intersects the ##x##-axis, like shown in the picture below:
If you cannot see them, then you may need to adjust your window. You can do this by pressing the “Window” key and adjusting the x-max and min to whatever you need.
Now you will need to find those intercepts. You can do this by hitting 2nd -> Calc (normally the “Trace” button) -> Zero. Once you have done this, the calculator will ask you to specify the left bound of the zero. To do this, move the cursor using the arrow keys (or enter an x-value in directly using the number keys) so that it is somewhere close to the left of where the function crosses the x-axis. Press ENTER.
Then, specify the right bound in the same way, except you want the cursor to be to the right of the crossing point.
Lastly, it will ask you to guess where the zero is. You can help the calculator find the zero by moving the cursor close to the crossing point. Then, the calculator will report the zero, which will be the x-value. If there are other real zeros, you can find them using this method.
In addition, whenever you find a zero your calculator automatically stores it as your “X” value. Therefore if you want to use that zero value for something else (plugging it into another equation, etc), then you can do that by just using the “X” key (next to Alpha) wherever you want that value.
Here is a video that might help you as well: