Re: Relationship between roots of CUBIC equation!
- From: "Nasser Abbasi" <nma@xxxxxxxxx>
- Date: Fri, 2 Mar 2007 21:35:16 -0800
"lancsru" <lancsru@xxxxxxxxx> wrote in message
news:1172877233.156608.25230@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Hello
Could anybody kindly tell me if there is any relationship between
roots of a CUBIC equation, ax^3+bx^2+cx+d=0, where a, b, c and d are
real numbers? Let the roots be r1, r2 and r3. For example, for the
quadratic equation ax^2+bx+c=0, where r1 and r2 are the roots. These
two roots have the properties: 1) r1+r2=-b/a 2) r1*r2=c/a
Do the roots of a CUBIC equation also have similiar properties?
Thanks a lot.
Mike
One relation that I know about relates the upper bound and the lower bound
on the size of the root. (abs value of roots).
The maximum root of a polynomial of degree n is bounded above by the
following
|largest root| = 1 + max( abs( a_i) )/ abs(a_n) where 'i' goes over
all the coefficients except a_n
and the minumum root is bounded below by
|smallest root| = 1/A
where A = 1 + max( abs( a_i) )/ abs(a_0) where 'i' is goes over all the
coefficients except a_0
for example, 3 x^3 + 2 x^2 + x + 5 = 0
then abs value of the largest root is less than 1 + max{2,1,5}/3 , i.e.
1+5/3 ===> less than 8/3=2.6666667
and abs value of the smallest root is larger than 1/A, where A=1 +
max{3,2,1}/5=1+3/5=8/5, hence smallest root is larger than 5/8=0.625
So given the above cubic, you know the region where all roots must be
located in the general complex plane.
Nasser
.
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