3.7 Solving Polynomial Equations

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Presentation transcript:

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

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

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

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.

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

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

-1 2 -5 1 4 -4 -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: -1 2 ?????