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Finding the Potential Zeros.  A Theorem that provides a complete list of possible Rational Roots or Zeroes of the Polynomial Equation.  A Root or Zero.

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Presentation on theme: "Finding the Potential Zeros.  A Theorem that provides a complete list of possible Rational Roots or Zeroes of the Polynomial Equation.  A Root or Zero."— Presentation transcript:

1 Finding the Potential Zeros

2  A Theorem that provides a complete list of possible Rational Roots or Zeroes of the Polynomial Equation.  A Root or Zero of a function is a number that, when plugged in for the variable, makes the function equal to zero.  It states: if P(x) is a polynomial with integer coefficients and if (p/x) is a zero of P(x).

3  P is a a factor of the constant term of P(x)  Q is a factor of leading coefficient of P(x)  EXAMPLE:  P(x)=2x 2 + x 3 – 19x 2 - 9x + 9  P is all factors of 9, which are: +/- 1, +/- 3, +/- 9  Q is all factors of 2, which are +/- 1, +/- 2

4  1. Arrange the forms of the polynomial in descending order by exponent.  2. Write down all factors of constant term. These are all possible values for P.  3. Write down all possible values of leading coefficient. These are all possible values for Q  4. Write down all possible values of P/Q, which equals all possible zeros.  5. Use your calculator to find all real zeroes  6.Use Synthetic or Long division to show which values for (p/q) will be a factor.

5  1. Press [y=] Plug your equation into your Y 1 =  2. Press [2 nd ] [graph]  3. Find any Zeroes under the “Y 1 ” Column  4. Look across to your “X” Column and match up the x that goes with the y=0  5. Which ever number your X is that matches up with your Y=0 is your real Zero.

6  *This is an example from before*  P(x)=2x 4 + x 3 – 19x 2 - 9x + 9 first find your P’s and Q’s (which we found before) P = +/- 1, +/- 3, +/- 9 Q= +/- 1, +/- 2 Now divide your P’s from your Q’s. P/Q ** Remember that (p/q) will be both negitive and positive. Simplify each and make sure there are no duplicates **

7  Your P/Q are:  +/- 1, +/- ½, +/- 3, +/- 3/2, +/- 9, +/- 9/2  12 possible zeros is your answer!  Now use Synthetic or Long division to determine which values for p/q will equal 0.

8  Plug 2x 4 + x 3 – 19x 2 - 9x + 9 into your Y 1  Press [2 nd ] [Graph]  Match up your Y 1 =0 to your X  Your Real Zeroes are :  - 3, - 1, & 3  **Notice how ½ didn’t appear on the calculator that’s why you should always do synthetic or long division and check with your calculator**

9

10  1. P(x)= x 3 -2x 2 -29x+30 ◦ Find P/Q  2. P(x)= x 4 -x 3 -11x 2 -x-12 ◦ Factor Completely

11  REAL ZEROS:  1. -5,1, & 6  2. -3, & 4

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