We all remember from high school that they are called quadratic equations and can be solved easily using the quadratic formula. Formally, a quadratic equation is a polynomial of second degree that has the form:
such that a =/= 0.
Similarly, we remember polynomials of three degrees, which are called cubic equations and follow the form ax^3 + bx^2 + cx + d. Clearly we all remember how to find the roots of these functions using various formulas. A polynomial of degree four is called a quartic function and it to can be solved in a relatively straightforward manner.
But a problem arises when trying to solve the dreaded quintic function, which has the form:
Quintic equations have graphs that have a shape something like this:
It is a something that is never mentioned back when we are first learning to solve polynomial equations, but there exists a theorem known as the Abel-Ruffini theorem (published in 1824) which states that there are no general solutions to polynomial equations of degree 5 or higher.
But not all hope was lost!
In 1885, a class of quintic functions were discovered which can be solved. These are irreducible quintics of the form:
where all coefficients are rational. This equation is solveable by radicals <=> it is in the form:
where μ and ν are rational.
To get an even better idea of how terribly convoluted the solutions are to quintic functions, realize that in order to properly solve these equations, one has to use the tools of linear algebra to construct a companion matrix and then find the eigenvalues of that matrix!
So next time you are walking down the street, finding the roots of a second, third, or fourth degree polynomial functions, realize how lucky you really are!! Take the time to appreciate the formulas that are in place for them!
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