Rational Root TheormFinding the Potential Zeros

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 Rational Root Therom?

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)=2x2 + x3 – 19x2 - 9x + 9 P is all factors of 9, which are: +/-1,+/- 3,+/- 9 Q is all factors of 2, which are +/- 1, +/- 2

What is P and Q?

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.

Steps on How to Find Your Zeroes!

1. Press [y=] Plug your equation into your Y1=

2. Press [2nd] [graph]

3. Find any Zeroes under the “Y1” 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.

How to Find Real Zeroes using your CALCULATOR!!

*This is an example from before*

P(x)=2x4 + x3 – 19x2 - 9x + 9 first find your P’s and Q’s (which we found before) P = +/-1,+/- 3,+/- 9Q=+/- 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 **

LETS DO AN EXAMPLE TOGETHER!

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.

Continuation of Example 1!

Plug 2x4 + x3 – 19x2 - 9x + 9 into your Y1

Press [2nd] [Graph]

Match up your Y1=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**

Graphing it On a Calculator!

Synthetic and Final Answer!

1. P(x)= x3-2x2-29x+30◦ Find P/Q

2. P(x)= x4-x3-11x2-x-12◦ Factor Completely

Here are More Examples to Try!

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

ANSWERS: (BASED ON CALCULATOR)