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6.6-6.7 Finding All Real Zeros

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