Rational Root Theorm Finding the Potential Zeros What

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Rational Root Theorm Finding the Potential Zeros

Rational Root Theorm Finding the Potential Zeros

What Is Rational Root Therom? �A Theorem that provides a complete list of possible

What Is Rational Root Therom? �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).

What is P and Q? �P is a a factor of the constant term

What is P and Q? �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)=2 x 2 + x 3 – 19 x 2 - 9 x + 9 �P is all factors of 9, which are: �Q is all factors of 2, which are +/-1, +/+/- 1, 3, +/- 9 +/- 2

Steps on How to Find Your Zeroes! � 1. Arrange the forms of the

Steps on How to Find Your Zeroes! � 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.

How to Find Real Zeroes using your CALCULATOR!! � 1. Press [y=] Plug your

How to Find Real Zeroes using your CALCULATOR!! � 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.

LETS DO AN EXAMPLE TOGETHER! � *This is an example from before* � P(x)=2

LETS DO AN EXAMPLE TOGETHER! � *This is an example from before* � P(x)=2 x 4 + x 3 – 19 x 2 - 9 x + 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 **

Continuation of Example 1! � Your P/Q are: � +/-1, +/- ½, +/-3/2, +/-9/2

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

Graphing it On a Calculator! � Plug 2 x 4 + x 3 –

Graphing it On a Calculator! � Plug 2 x 4 + x 3 – 19 x 2 - 9 x + 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**

Synthetic and Final Answer!

Synthetic and Final Answer!

Here are More Examples to Try! � 1. P(x)= x 3 -2 x 2

Here are More Examples to Try! � 1. P(x)= x 3 -2 x 2 -29 x+30 � 2. P(x)= x 4 -x 3 -11 x 2 -x-12 ◦ Find P/Q ◦ Factor Completely

ANSWERS: (BASED ON CALCULATOR) � REAL ZEROS: � 1. -5, 1, & 6 �

ANSWERS: (BASED ON CALCULATOR) � REAL ZEROS: � 1. -5, 1, & 6 � 2. -3, & 4