Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Manipulation of (Imaginary?) Roots


Date: 8/18/96 at 19:46:41
From: Anonymous
Subject: Manipulation of (Imaginary?) Roots

I would appreciate it if you could point out the mistake I'm making.

Let r,s, and t be the roots of x^3-6x^2+5x-7=0. 
Find 1/r^2+1/s^2+1/t^2.  

Simplifying the fractions into one, I get 
[(st)^2+(rt)^2+(sr)^2]/(rst)^2    

Let the numerator be x.
Expand (rs+st+rt)^2 = x+2rst(r+s+t)

Using the coefficients from the original equation I get 
25 = x+2*7*6  or  x = -59 and the desired fraction is -59/49. 

The problem is that everything is squared in the desired fractions, 
so they cannot be negative.  Two of the roots are imaginary, but I 
don't think that should affect the result.

Thanks,
   Michael


Date: 8/19/96 at 4:34:42
From: Doctor Pete
Subject: Re: Manipulation of (Imaginary?) Roots

The answer you obtained is indeed correct.  In fact, it is precisely 
because two of the roots are imaginary that the above result is 
possible!

To see why this is so, take a look at the original expression,

     1/r^2 + 1/s^2 + 1/t^2 .

If, say, r and s were imaginary, r^2 and s^2 would be real and 
*negative*. Thus 1/r^2 and 1/s^2 would be negative, and one could 
easily expect the entire sum to be negative as well.  In this case, 
the approximate numerical values of the three roots are

     {5.30633, 0.346833 + 1.09494 I, 0.346833 - 1.09494 I}

Since the squares of complex conjugates are also complex conjugates, 
it is clear that the imaginary component vanishes when the sum of the 
reciprocals is taken.  Since the real root is relatively large, the 
reciprocal of its square will be very small - small enough that the 
net result is negative.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Basic Algebra
High School Imaginary/Complex Numbers

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/