Polynomials are a fairly straightforward expression in mathematics, containing only x each raised to a positive whole number power:
Where the highest power n is called the degree of the polynomial.
Everyone that has a taste of serious mathematics, especially algebra, will at some point encountered the polynomials not only because of its simplicity, and also it serves as a very important stepping stone from the simple linearity into the more complex non-linear areas in mathematics.
Solving low degree polynomial equations
Of course we will like to know if given an equation involving a polynomial where P(x)=0, can we solve it? Fortunately we can do it easily with degree 1 and 2, as they are nothing more than the linear and quadratic equations respectively.
For linear equations (degree 1 polynomial), we can solve it pretty easily:
For quadratic equations (degree 2 polynomial), things get a bit more complicated but we can still solve it fairly easily by the famous quadratic formula:
Can we do the same for cubic equations, which are degree 3 polynomials? Yes, as one of their solution can be obtained by the very complicated cubic formula (obtained from http://www.math.vanderbilt.edu/~schectex/courses/cubic/):
Where the other solution can be obtained by a slight modification of the formula which I'm not going to get into here. Now you see why they don't teach this formula in high schools (or even in uni for that matter). The presence of cube roots is not surprising, as a cubic equation involves a cube term.
What about quartic equations then? It is possible to express their solution as a formula, the so called quartic formula, but it is a very very complicated equation which I will not include here. (You can check out the formula from wikipedia if you're interested here). Also note that the formula involves a forth root, which once again is expected.
Naturally we'd like to ask, is it possible to express the solution of a quintic equation (polynomial of degree 5) using a formula like we did for linear, quadratic, cubic and quartic equations?
Unfortunately this where the solvability of polynomial breaks down, as according to Abel-Ruffini's theorem: It is impossible to express the solution of a polynomial equation with degree 5 or higher as a formula involving only arithmetic (addition, subtraction, multiplication and division) and roots (also known as radicals).
This means that the arithmetic and root operations are not enough for us to find out what the solutions are. This is not to say polynomials of degree 5 or higher are unsolvable, it can be solved be it needs more than the arithmetic and root operations, which means its formula will be way more complicated than the quartic formula.
The proof for Abel-Ruffini's theorem involves the use very advanced area in mathematics called Galois Theory, which is the realm of one of the purest form of mathematics that is way beyond me, so of course I will not write them here.
Of course no such formula have been discovered yet, apart from a few special cases, and even if it is discovered, I highly doubt it will even be useful.
Can polynomials be solved then?
The good news is that even though we cannot generally express solutions to polynomial equations exactly or simply, they can still be solved very easily numerically. Meaning you can get a very good numerical approximation of the solution easily, which in practice is all you need anyway. For example, the value of pi has an approximate value of 3.14159.
There a tonnes of method to solve equations numerically, collectively known as root-finding algorithms. The study of such methods are an area of mathematics called numerical analysis, which are used heavily in sciences and employed by calculators.
One of such method is called Newton's Method, which is my personal favorite to use when finding an approximate solution to a equation, but I'm getting carried away here so I'll save this till next time!