All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
How many roots? (Posted on 2011-07-31) Difficulty: 3 of 5
Prove that the equation
x^4-4*x-1=0
has at most 2 real number solutions.

No Solution Yet Submitted by Ady TZIDON    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution (spoiler) - I think | Comment 1 of 6
The roots of a polynomial always come in complex conjugate pairs.  Therefore there are zero, two or four real roots.  Assume the general case that the roots are a+-bi and c+-di.  The product of all the roots must equal -1.  Therefore -1 = a^2 + b^2 +c^2 +d^2.  All real numbers have a square>=0, therefore at least one of the terms a,b,c or d must be imaginary, and therefore at least 2 must be imaginary.  Therefore there can be a maximum of only 2 real roots.

  Posted by Kenny M on 2011-08-02 17:48:55
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information