1. Warm up Use Synthetic Division: 1. 3x 2 – 11x + 5 x – 4 2. 5x 5 + 3x 3 +1 x + 2

2. Lesson 4-6 Rational Root Theorem Objective: To use the rational root theorem to determine the number of possible rational roots in a polynomial.

3. Rational Roots Theorem: If a polynomial equation has a rational root, then this root is one of the possible quotients of a factor of the constant term, divided by a factor of the leading coefficient.

4. Ex. List the possible rational roots for the following polynomials. factors of constant: factors of lead coefficient: possible rational roots:

5. factors of constant: factors of lead coefficient: possible rational roots: Put in order first:

6. Let’s Try One Find the POSSIBLE roots of 5x 3 -24x 2 +41x-20=0

7. Let’s Try One 5x 3 -24x 2 +41x-20=0

8. That’s a lot of answers! Obviously 5x 3 -24x 2 +41x-20=0 does not have all of those roots as answers. Remember: these are only POSSIBLE roots. We take these roots and figure out what answers actually WORK.

9. Step 1 – find p and q p = -3 q = 1 Step 2 – by RRT, the only rational root is of the form… Factors of p Factors of q

10. Step 3 – factors Factors of -3 = ±3, ±1 Factors of 1 = ± 1 Step 4 – possible roots -3, 3, 1, and -1

11. Step 5 – Test each root Step 6 – synthetic division X X³ + X² – 3x – 3 -3 3 1 (-3)³ + (-3)² – 3(-3) – 3 = -12 (3)³ + (3)² – 3(3) – 3 = 24 (1)³ + (1)² – 3(1) – 3 = -4 (-1)³ + (-1)² – 3(-1) – 3 = 0 THIS IS YOUR ROOT BECAUSE WE ARE LOOKING FOR WHAT ROOTS WILL MAKE THE EQUATION =0 1 1 -3 -3 0 1-3 3 0 -1 0 1x² + 0x -3

12. Step 7 – Rewrite x³ + x² - 3x - 3 = (x + 1)(x² – 3) Step 8– factor more and solve (x + 1)(x² – 3) (x + 1)(x – √3)(x + √3) Roots are -1, ± √3

13. Sources Ponderosa High School Math Department. Ponderosa High School, n.d. Web. 21 Jan. 2013.. "6.5 Theorems About Roots of Polynomial Equations." Pleasanton Unified School District. N.p., n.d. Web. 21 Jan. 2013..