# Just Stuff

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## 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 $n$ so the Cauchy lower bound is: If  $x$ is a root of the polynomial:

$P(x)=a_nx^n+a_{n-1}x^{n-1}+ ... + a_0$

then:

$|x| \le 1+ \frac{\max_{i=0, .. , n-1} |a_i|}{|a_n| }$

Now I will extend this to give a lower bound for the absolute value of the non-zero roots of $P(x)$ Without any real loss of generality we can assume that both $a_n$ and $a_0 \ne 0$ (if the lowest power of  $x$ appearing in $p(x)$ is $\rho \ne 0$ we divide through by $x^{\rho}$ to get a polynomial of the required form and whos non-zero roots are the same as the polynomial we started with).

Let $B(a_0, a_1, ..., a_n)$ be an an upper bound on the ansolute value of roota of the ploynomial: $P(x)=a_nx^n+a_{n-1}x^{n-1}+ ... + a_0$ where $a_n$ and $a_0$ are both non-zero.  Now let $x$ be a root of $P(x)$, so:

$P(x)=a_nx^n+a_{n-1}x^{n-1}+ ... + a_0=0$

Now divide through by $x^n$ and put $y=1/x$, so we have:

$Q(y)=a_0 y^n + a_1 y^{n-1}+ ... + a_n=0$

Hence:

$|y| \le B(a_n, a_1, ..., a_0)$

or:

$|x| \ge 1/B(a_n, a_1, ..., a_0)$.

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

$\frac{1}{B(a_n, a_{n-1}, ..., a_0) } \le |x| \le B(a_0, a_1, ..., a_n)$

and for the Cauchy bound we have:

$B(a_0, a_1, ... , a_n) =1+\frac{\max_{i=0, .. , n-1} |a_i|}{|a_n| }$

So:

$\frac{|a_0|}{|a_0|+\max_{i=1,..,n}|a_i|}\le |x| \le \frac{|a_n|+\max_{i=0,..,n-1}|a_i|}{|a_n|}$

References: