<s>[INST] When, where and **how often** do you find polynomials of higher degrees than two in mathematical, pure/applied, research? A formula for solving a polynomial of degree three, see this link; $ax^3+bx^2+cx+d=0$, is
$$\begin{align}
x\quad&=\quad
      \sqrt[3]{
          \left(
              \frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a}
          \right) 
          +      
          \sqrt{
              \left(
                  \frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a}
              \right) ^2  
              + 
              \left(
                  \frac{c}{3a} - \frac{b^2}{9a^2}
              \right) ^3 
          }
      }\\
      &+\quad
      \sqrt[3]{
          \left(
              \frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a}
          \right) 
          -      
          \sqrt{
              \left(
                  \frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a}
              \right) ^2  
              + 
              \left(
                  \frac{c}{3a} - \frac{b^2}{9a^2}
              \right) ^3 
          }
      }
      \;-\;\frac{b}{3a}
\end{align}$$

Unlike quadratic, cubic, and quartic polynomials, the general quintic
  cannot be solved algebraically in terms of a finite number of
  additions, subtractions, multiplications, divisions, and root
  extractions, as rigorously demonstrated by Abel (Abel's impossibility
  theorem) and Galois. However, certain classes of quintic equations can
  be solved [...] Source: http://mathworld.wolfram.com/QuinticEquation.html

At levels of $5^{\text{th}}$ degree polynomials, things are starting to look really serious in my eyes. My question is:

If it is possible to not answer subjectively: When, where and how often do you find polynomials of higher degrees than two in mathematical, pure/applied, research? 

 [/INST] Gaussian quadrature is a class of methods of numerical integration based on polynomial approximation. Typically, polynomials of degree $5$ or $10$ are employed. For many of the functions that arise in applied mathematics, these give very accurate results. Polynomials of arbitrarily high degree can in principle be used, but the additional accuracy is seldom relevant or worth the extra complexity entailed. 
</s>