1. 3.7 Solving Polynomial Equations That is, finding all the roots of P(x) without a head start

2. Example: Prime factor 10406 What is your process? Why?

3. There are 5 main rules we will use to determine possible rational roots. There are others that you can read about in the book, but these 5 are the basic ones you narrow down the possibilities. Remember: when you divide synthetically, if the remainder = 0 then the number is a root. If the remainder ≠0 then the number is not a root and never will be.

4. Rule #1 The only possible real rational roots are Where

5. Rule #2 If the signs of the all the terms in the polynomial are +, then all roots are negative. Think about this, using

6. Rule #3 If the signs of the terms of the polynomial alternate 1 to 1 (that is + – + – + –) then all the roots are positive. If a term is missing, it is ok to assign it a + or – value to make it fit this rule.

7. Rule #4 If you add all the coefficients and get 0, then 1 is a root. Otherwise 1 is not a root (and never will be a root ever). This is a good one. Essentially if it works, you have your start point.

8. Rule #5 Change the signs of the odd powered coefficients and then add. If you get 0, then -1 is a root. Otherwise, -1 is not a root. Sometimes this one isn’t worth the effort.

9. Why these rules? You will have a list of possibilities (and maybe a definite) with which to start synthetically dividing. Remember – the goal of the problem is to find all zeroes (or factor). A zero is something whose factor divides evenly into a function. Therefore, synthetically you want to get a remainder of 0.

10. Find zeroes, then factor the following.