Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Precalculus Mathematics Homework Help
To find the nature of roots of a quintic equation....
Reply to thread
Message
[QUOTE="epenguin, post: 6031182, member: 106258"] You could reflect on the derivative of your polynomial. You can actually solve that – how many real zeros does it have? What does that imply for the real zeros of the original polynomial? (In prospective, the general and powerful algebraic method which can tell you the number of real roots of an algebraic equation in one variable that lie between two numbers, a and b say, even when you don't or can't solve it, is called Sturm's method. With it you can as special case tell the total number of real roots, a and b are simply + and - infinity. It is, as I say, powerful, and you should get onto it before too long, but on the other hand algebraists are very fond of doing no more than necessary. And this full power is not necessary to answer your question. It was not necessary to use it for example to find, as you have, that there is one positive root, a more limited method was sufficient . There is a 'limited' theorem in which answers your question called De Gua's rule. I quote (paraphrase) from my source for it which is "The Theory of Equations" by Burnside and Panton (publ 1904! I think you can find it online). [I]"When 2m successive terms of an equation are absent, the equation has 2m nonreal complex roots; and when 2m + 1 successive terms are absent, the equation has 2m +2, or 2m non-real roots, according as the two-terms between which the deficiency occurs have like or unlike signs".[/I] Result - your equation has four non-real roots. Burnside and Panton derive De Gua's rule from another "limited" theorem called Fourier and Budan's theorem. But actually if you did the exercise as I recommend above, you could probably prove De Gua's rule yourself.) [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Precalculus Mathematics Homework Help
To find the nature of roots of a quintic equation....
Back
Top