1. Sec. 6-5: Theorems About Roots of Polynomial Equations

2. 1.Rational Root Theorem: A theorem which allows us to determine POSSIBLE RATIONAL roots of a polynomial equation by simply finding all FACTORS of : Constant Term Lead coefficient *** Look at p. 329’s definition. *** IRRATIONAL & Imaginary roots occur in PAIRS

3. ** Keep in mind: The degree of a polynomial determines the number of roots. So, if a CUBIC equation only has 1 RATIONAL root, then there must be 2 IRRATIONAL roots. Procedure: 1.List ALL factors of the constant and lead coefficients. 2. Substitute EACH value into the equation to see if you get “0”…if so, it is a root.

4. Let the Calculator do the work 1.Type equation in y= screen. 2.In “home” screen, STO one possible root into “x”. 3.Hit the VARS key & select y-vars, select FUNCTION and pick the appropriate y equation (usually #1) and hit ENTER. 4.If this result is ZERO, you’ve found a rational root, because of the remainder theorem.

5. Find the RATIONAL roots for: 3x 3 – x 2 – 15x + 5 = 0 1.Find all factors of 5/3: ±1, ±5 ±1, ±3 ±1, ±1/3, ±5, ±5/3 Now, use the calculator to determine that 1/3 is the ONLY rational root.

6. Now, we could use synthetic division by 1/3 to acquire a QUADRATIC equation that we could solve.