Bounds on roots of polynomials

What bound is there on the roots of a given polynomial, in terms of its degree and coefficients?

Consider the polynomial $p(x) = 3x^7 – 5x^3 + 42$. Would you not agree, without doing any calculation, that one million ($10^6$) cannot be a root? It just wouldn’t be in accord with the smallness of the coefficients and the well-behavedness of polynomials. And yet I don’t recall ever having encountered anything in the literature that gave a bound on the absolute value of the roots of a polynomial in terms of the degree and coefficients of the polynomial, but I’m pretty sure such must exist, and that I simply missed it, and so I’m tagging this a a reference-request.

By the way, during my post of a question, every time, it seems, there are stray graphics on the screen, as if from someone else’s question or answer. Is this happening to anyone else?

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Here’s an elementary bound. Let $p(x) = x^n – a_{n-1} x^{n-1} – … – a_0$. If $|x| \ge 1$ then $m \ge n$ implies $|x|^m \ge |x|^n$, hence if $p(x) = 0$ then

$$|x|^n = \left| a_{n-1} x^{n-1} + … + a_0 \right| \le |x|^{n-1} \left( |a_{n-1}| + … + |a_0| \right)$$


$$|x| \le \text{max}(1, |a_{n-1}| + … + |a_0|).$$

In your case we can do much better because of the zero coefficients: we get

$$|x|^7 \le \text{max}(1, \frac{47}{3} |x|^3)$$

hence $|x|^4 \le \frac{47}{3} < 16$, or $|x| < 2$.

A technique useful in special cases is the Gauss-Lucas theorem, and a technique useful in general cases is Rouche’s theorem.