1. Rational Root Theorem

2. Warm-Up

3. What happens when you aren’t given a zero or factor? If you are just given a polynomial that you can’t factor like in the warm-up, then you can use the Rational Root Theorem to identify… POSSIBLE RATIONAL ROOTS!

4. Let’s try the warm-up again without the given zero…

5. How did we come up with these numbers? To use the Rational Root Theorem, you have to look at the ratio of all of the factors of the constant term and the leading coefficient. The ± is included because the product of two negatives is a positive. EX: -1-30 = 30

6. Now, what do we do with these numbers? Since these are possible rational roots, we can use synthetic or polynomial long division to test them. **This part can be a little tedious, especially if you have a lot of possible zeros…Stay Strong! Test +1  1 1 0 -31 30 1 1 1 1 -30 0 This means that x = 1 is a root, and (x – 1) is a factor, therefor…

7. Once you get to this point, you can solve the quadratic using one of the tools in our toolbox: Factoring Quadratic Formula Completing the Square Remember, there is no quadratic that we cannot solve!

8. Let’s try another one… You don’t have to list repeats multiple times. These are the numbers we can test using synthetic division.

9. Step 2: Test Possible Roots Test + 1  1 2 -3 -3 2 Test – 1  –1 2 -3 -3 2 Possible Roots 2 2 -4 -2 This means that x = 1 is NOT a root, and (x - 1) is NOT a factor…try again! 2 -2 -5 5 2 -2 0 This means that x = -1 is a root, and (x + 1) is a factor, therefor… Try to avoid testing the fractions if possible. I will show you why.

10. Once again, at this point we just solve the quadratic by factoring, quadratic formula, or completing the square to find the other roots. **NOTE: For higher degree polynomials, such as quartics, quintics, and above, you repeat the Rational Root Theorem to test other possible rational roots. Don’t repeat on the original polynomial though. Apply the Rational Root Theorem to the new polynomial you got from the previous rational root test!