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

2. 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.

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

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

5. 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.

6. 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.

7. 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.

8. 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.

9. -1 1 3 3 1 -1 -2 -1 1 2 1 All Roots Negative (RULE 2)
-1 is a root (RULE 5) -1 -2 -1 1 2 1

10. RULE #5 Works: -1 is a root
Now do Synthetic division

11. -2 7 4 -8 2 -7 8 -4
Rule 3: Positive Root 2 4 -6 4 -3 2 2
Not factorable
Plug into quad to get the root
Roots: ?????