Advertisement Upgrade to remove ads

These flash cards use the rational zero theorem and the fundamental theorem of algebra to find ALL the zeros of a polynomial function.

Finding All Real Zeros: Step One

1. Use the polynomial function f(x) =3x³-8x² +5x-2 as an example. First look at the degree of the polynomial; (it is 3 so there are exactly 3 zeros or roots for this function). They may be real or complex zeros.

Finding All Real Zeros: Step Two

First find the factors of the constant term 2 which are 1, 2.

Then find the factors of the leading coefficient 3 which are 1 and 3.

The POSSIBLE rational roots for a positive or negative ± root is ± 1, ± 2, ±1/3, and ±2/3.

Finding All Real Zeros: Step Three

To find an actual root of the example, CHOOSE a rational root from the list in Step 3 and then use synthetic division (check your notes or page 330).

You get that 2 is the only rational root or zero and that (x-2) is a factor. Then the polynomial factors into (x-2) X (3x²-2x+1).

What is a ZERO of a polynomial function?

It's where the function crosses the x axis. This is also called the x-intercept(s).

Do ALL polynomial functions have ZEROS?

NO. For example, a parabola whose vertex is above the x-axis and opens upward, would have NO ZEROS because it never crosses the x-axis.

The graph of any function that does NOT cross the x-axis would have NO zeros.

Please allow access to your computer’s microphone to use Voice Recording.

Having trouble? Click here for help.

We can’t access your microphone!

Click the icon above to update your browser permissions above and try again


Reload the page to try again!


Press Cmd-0 to reset your zoom

Press Ctrl-0 to reset your zoom

It looks like your browser might be zoomed in or out. Your browser needs to be zoomed to a normal size to record audio.

Please upgrade Flash or install Chrome
to use Voice Recording.

For more help, see our troubleshooting page.

Your microphone is muted

For help fixing this issue, see this FAQ.

Star this term

You can study starred terms together

NEW! Voice Recording

Create Set