3.7 Solving Polynomial Equations

That is, finding all the roots of P(x)

without a head start

Example: Prime factor 10406

What is your process?

Why?

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.

Rule #1

The only possible real rational roots are

Where

pq

p factors of the constant

q factors of the high power coefficient

Rule #2

If the signs of the all the terms in the polynomial are +, then all roots are negative.

Think about this, using 2P(x) 2x 7x 3

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.

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.

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.

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.

Find zeroes, then factor the following.3 21. F(x) x 3x 3x 1

3 22. F(x) 6x 25x 21x 10

4 3 23. F(x) 2x 5x x 4x 4