Rational Root Theorem

Polynomials Unit
Rational Root Theorem (finding all rational factors of a polynomial)

1. Rational Root Theorem Unit 6: Polynomials

2. Finding All Factors

3. Recap We can use the Remainder & Factor Theorems to determine if a given linear binomial is a factor of a polynomial Remember: is a factor of if and only if . In other words, the remainder after synthetic division must be zero in order for the linear binomial to be a factor of the polynomial. Example: Prove is a factor of  

4. How do we find the other factors? The quotient we get after synthetic division is called the depressed polynomial We can FACTOR this depressed polynomial!!! Factor this polynomial using a previously learned method! GCF Simple Case (multiply to “c” & add to “b”) Slide & Divide

5. Example We have already proved is a factor of We can find the depressed polynomial from synthetic division . The depressed polynomial (quotient) is  

6. Example (Cont.) Now, lets factor our depressed polynomial using Slide & Divide…  

7. Example (Cont.) Therefore, all of the factors of our original trinomial are:  

8. Try #’s 1-4 from your worksheet on your own! PRACTICE

9. What happens when you must factor a polynomial of degree ≥ 3 and you do not know any factors?! Rational Root Theorem

10. Rational Root Theorem If has integer coefficients, then every rational zero of has the following form:  

11. Example Find all possible rational roots of using the Rational Root Theorem Factors of the constant term: Factors of the leading coefficient: Possible r ational zeros: Simplified list:  

12. Try #’s 5-8 from your worksheet on your own! PRACTICE

13. Example 1 Find all the rational roots of the given function Possible Zeros: Check using Remainder & Factor Theorems: Since 2 gives a 0 remainder, that means is a factor.  

14. Example 1 (Cont.) Now, use synthetic division to find the depressed polynomial Factor to find the remaining factors Therefore, all the factors are:  

15. Example 2 Find all the rational roots of the given function Notice that this polynomial has a GCF of x! Factor out the GCF: Possible Zeros: Check using Remainder & Factor Theorems:  

16. Example 2 (Cont.) Since gives a 0 remainder, that means is a factor. Now, use synthetic division to find the depressed polynomial Factor to find the remaining factors Therefore, all the factors are:  

17. Try #’s 9-10 from your worksheet on your own! PRACTICE