A natural question to ask when given a polynomial is: how do the roots and the coefficients interplay with one another? In fact a huge portion of polynomial theory and solving algebraic equations is finding out the details of this relationship. The quadratic formula is a prime example – It gives a way of finding roots by only using the values of the coefficients. An explicit relationship, an algebraic formula, is only possible for polynomials of degree less than which is a famous result pioneered by the work of Galois.
The interplay we are going to determine for polynomials of any degree is that you only need to look for roots in a neighborhood around with the neighborhood’s size depending only on the size of the polynomial’s coefficients.
Let be a polynomial over of degree with the representation where and that for all and some . We will be considering our polynomial as a function and be setting and where our choice of will be more apparent later.
What we are going to show is that for of a large enough magnitude our polynomial will have a magnitude as large as we like. We will also establish a simple bound on the magnitude of the roots of our polynomial .
Suppose , we have by the definition of our constants and estimating gives by using the sum of a geometric series and the triangle inequality.
Using the triangle inequality allows us to conclude which gives us by using the previous estimation as both and as .
Bringing this together we have shown for all if we look at where we have .
Looking at the contrapositive of this statement lets us conclude if for some and we must have or equivalently living in the ball . If we take to be a root of then by setting we have that giving us a bound on the size of our root in terms of the polynomials coefficients. This means all roots of will be found in the ball .