1. Roots & Zeros of Polynomials III
Using the Rational Root Theorem to Predict the Rational Roots of a Polynomial
Created by K. Chiodo, HCPS

2. Find the Roots of a Polynomial
For higher degree polynomials, finding the complex roots (real and imaginary) is easier if we know one of the roots.
Descartes’ Rule of Signs can help get you started. Complete the table below:

3. The Rational Root Theorem
The Rational Root Theorem gives us a tool to predict the values of rational roots:

4. List the Possible Rational Roots
For the polynomial:
All possible values of:
All possible rational roots of the form p/q:

5. Narrow the List of Possible Roots
For the polynomial:
Descartes’ Rule:
All possible Rational Roots of the form p/q:

6. Find a Root That Works For the polynomial:
Substitute each of our possible rational roots into f(x). If f(a) = 0, then a is a root of f(x). (Roots are the solutions to the polynomial set equal to zero!)

7. Find the Other Roots
Now that we know one root is x = 3, do the other two roots have to be imaginary? What other category have we left out?
To find the other roots, divide the root we know into the original polynomial:

8. Find the Other Roots (con’t)
The degree of the resulting polynomial is 1 less than the original polynomial. When the resulting polynomial is a QUADRATIC, we can solve it by FACTORING or by using the QUADRATIC FORMULA!

9. Find the Other Roots (con’t)
This quadratic does not have real factors, but it can be solved easily by moving the 5 to the other side of the equation.

10. Find the Other Roots (con’t)
The roots of the polynomial equation:
are:

11. Another Example Descartes’ rule of signs: # Total Roots = 4
# + Real Roots = 1
# - Real Roots = 3 or 1
# Imag. Roots = 0 or 2
The possible RATIONAL roots:

12. Another Example Find a Rational Root that works:
f(2) = 0, so x = 2 is a root
Synthetic Division with the root, x = 2:

13. Another Example The reduced polynomial is:
Since the degree > 2, we must do synthetic division again, go back to the list of possible roots and try them in the REDUCED polynomial. The same root might work again - so try it also!
f(-1/3) = 0, so x = - 1/3 is a root

14. Another Example
Synthetic Division with the root, x = - 1/3 into the REDUCED POLYNOMIAL:
The reduced polynomial is now a quadratic:

15. Another Example Solve the resulting quadratic using the quad. formula:
The 4 roots of the polynomial are:

16. More Practice
For each of the polynomials on the next page, find the roots of the polynomials.
Know what to expect ---- Make a table using Descartes’ Rule of Signs.
List the possible RATIONAL roots, p/q.
Find one number from your p/q list that makes the polynomial = 0. You can check this either by evaluating f(#) or by synthetic division.
Do synthetic division with a root - solve the resulting polynomial.

17. More Practice
Find the roots of the polynomials: