## More on Bounds on the Roots of Polynomials

In an earlier article/post (reference 1) I showed the Cauchy upper bound for the absolute cvalue of the roots of a monic polynomial. The restriction to a monic polynomial was purly for convienience, but here we will require an arbitary polynomial of degree so the Cauchy lower bound is: If is a root of the polynomial:

then:

Now I will extend this to give a lower bound for the absolute value of the non-zero roots of Without any real loss of generality we can assume that both and (if the lowest power of appearing in is we divide through by to get a polynomial of the required form and whos non-zero roots are the same as the polynomial we started with).

Let be an an upper bound on the ansolute value of roota of the ploynomial: where and are both non-zero. Now let be a root of , so:

Now divide through by and put , so we have:

Hence:

or:

.

So combinining this lower bound with the upper bound we have for any -th degree polynomial with only non-zero roots for any root :

and for the Cauchy bound we have:

So:

♦

References:

1. https://captainblack.wordpress.com/2009/03/08/cauchys-upper-bound-for-the-roots-of-a-polynomial/

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