6.6-6.7 Finding All Real Zeros
|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.