A General Way of Finding the Zeroes of a Polynomial

As all of us know, the easiest method currently available in finding the zeroes of a polynomial is based in factorisation of polynomials.

I tried to find a faster way of calculating the zeroes of a quadratic polynomial, but ended up getting a trivial rewrite of the quadratic formula :

If $f(x) = ax^2+bx+c$, then the zeroes of the polynomial $f(x) = \frac{-b}{2a} \pm \sqrt{f(\frac{-b}{2a})\times \frac{-1}{a}} $

Looking at a linear polynomial $ax + b$, $x = \frac{-b}{a}$ is its zero.

Observing the above forms, we can see that the denominator of the zeroes get multiplied by the polynomial's degree (in both the polynomials) and an extra term comes where the multiplicative inverse (reciprocal) of the degree is its power (as in the case of the quadratic polynomial).

Now, my doubt is : can the zeroes of any polynomial be found using such forms (as given above)? Maybe some $\frac{-b}{na} (n = \text{degree of the polynomial})$ can be used to deduce the zeroes faster ? Or are all the formulae to find the zeroes of a polynomial of a specific degree (be it $2,3,4,5.... $etc.) based on such forms ?

I have only heard of the ways to calculate the zeroes of the polynomials of degree $1$ to $3$ and nothing more, since I am a tenth grader. That's why I am asking this.

Solution 1:

Now, my doubt is : can the zeroes of any polynomial be found using such forms (as given above)?

You are asking a very deep question - one of the main problems of mathematics in the 18th and 19th century was to figure out if the roots of any given polynomial can be calculated by means of a formula (or formulas) that depends only on the coefficients of a polynomial and uses only the operations of multiplication, division, addition, subtraction and taking any $n$-th root. We say that such formulas are in terms of radicals.

It was known that for polynomials of degree $1$, $2$, $3$ and $4$ formulas in terms of radicals exists, and you may look them up on e.g. Wikipedia. The formula for degree $4$ polynomials is especially elaborate.

However, for degree $5$ and higher such a formula in terms of radicals was not known, and in fact it was proven by Ruffini, and later Abel, that such a formula does not exist.

A bit later Galois proved the same fact in a very elegant way, creating in the process a theory that bears his name, Galois Theory, that 200 years later is still a very active research area, unfortunately beyond the reach of 10th grade mathematics.

However, if you are interested in a proof of this fact, there is a textbook aimed at high-school students in which the proof is presented through a series of problems that introduce basic group and field theory and complex analysis. The book is called

Abel’s Theorem in Problems and Solutions

by V.B. Alekseev (it's maybe worthwhile to note that the book is based on the lectures of V. Arnold who delivered them at one of the Moscow State Schools specializing in mathematics, so it will have a heavier maths course load than a typical American high school, though the material is still very accessible).

Solution 2:

Now, my doubt is : can the zeroes of any polynomial be found using such forms (as given above)? Maybe some $\frac{-b}{na} (n = \text{degree of the polynomial})$ can be used to deduce the zeroes faster ?

Kind of, yes!

As you know, the quadratic formula can be used to find the roots of any quadratic polynomial. And as you've noticed, the quadratic formula (as you've written it) consists of the term $\frac{-b}{na}$, where $n$ is the degree of the polynomial (which is 2), plus some variable stuff. (By "variable stuff," I mean the part that looks like $\pm \sqrt{\text{something}}$, where choosing a plus sign gives you one root and choosing a minus sign gives you the other one.)

Does the pattern continue? Yes, it does! There is also such a thing as the cubic formula, which can be used to find the roots of any polynomial of degree 3. The cubic formula is somewhat complicated, but it does in fact consist of the term $\frac{-b}{3a}$ plus some "variable stuff," just like the quadratic formula.

After that, there's the quartic formula, which is even more complicated, but which can be used to find the roots of almost any polynomial of degree 4. Just as you would expect, the quartic formula consists of $\frac{-b}{4a}$ plus some "variable stuff".

Unfortunately, there's no formula which can be used for finding the roots of any polynomial of degree 5—at least, there isn't one that uses only the "ordinary" kinds of functions that are used.

Nevertheless, the pattern does continue. The average of all of the roots of a polynomial is always $\frac{-b}{na}$, where $n$ is the degree of the polynomial, $a$ is the coefficient of $x^n$, and $b$ is the coefficient of $x^{n-1}$. (I actually didn't know this before—you've taught me something!)

Here is the proof. Any polynomial can be written as $a(x - r_1)(x - r_2)\cdots(x - r_n)$, where $n$ is the degree of the polynomial. If we multiply this out, it's possible to see that $b$ (that is, the coefficient of $x^{n-1}$) is $-a$ times the sum of all of the roots. As an equation:

$$b = -a \cdot (\text{sum of roots}).$$

By dividing both sides by $-na$, we see that

$$\frac{-b}{na} = \frac{\text{sum of roots}}{n}.$$

The right-hand side is, of course, the average of all of the roots.