A polynomial is an expression of the form \[\sum_{k=0}^n a_{n-k}x^k,a\neq0\] with coefficients defined over a field (such as \[\mathbb{R}\], \[\mathbb{C}\] or \[GF(p^k)\]). Polynomial algebra itself forms a ring. In a nutshell, it's a sum of powers of \(x\) where every power has is multiplied by a number coefficient 

\[P = a_0 x^n + a_1 x^{n-1} +\cdots + a_{n-1}x + a_n\]

Solving polynomial equations means finding values of \[x\] such that when the value is substituted the polynomial evaluates to zero
\[\sum_{k=0}^n a_{n-k}x^k = 0\]

Second degree polynomial equation \[a_0x^2+a_1x + a_2=0\] is called a quadratic equation and can be solved using a quadratic formula

\[x_{1,2} = \frac{-a_1\pm\sqrt{a_1^2 - 4a_0a_2}}{2a_0} 
\]
Cubic and quartic formulas also exist but they are much more involved while formulas for degree five or higher have been proven not to exist by the Abel-Ruffini theorem.

Solutions to a polynomial equation are called roots, which may or may not exist depending on the polynomial equation and field the polynomial is defined over, but according to the fundamental theorem of algebra solutions always exist for polynomial equations over the field of complex numbers \[\mathbb{C}\]. Knowing roots allows us to factor a polynomial: 
 \[\sum_{k=0}^n a_{n-k}x^k = a_0\prod_{k=0}^n(x-x_k)\] 
Polynomial coefficients and roots are also linked via Vieta's formulas.
 
Generally finding solutions to or factoring a polynomial is an NP-hard problem (hard even for computers) but, of course, numeric approximation methods exist (like Newton's method).

While finding solutions to a polynomial equation is hard, evaluating a polynomial for any given \(x\) is extremely easy (it's just multiplication and addition). Many functions can be represented by a polynomial via Taylor series (or a special case called Maclaurin series) and thus be computed easily by a computer or by hand with as much precision as you'd like. For example:
\[\sin x = \sum_{k=0}^{\infty}\frac{(-1)^kx^{2k+1}}{(2k + 1)!} = x - \frac{x^3}{3\cdot 2\cdot 1} + \frac{x^5}{5\cdot 4 \cdot 3\cdot 2\cdot 1}-\dots\]
For small \[x\], \[\sin x\approx x\] is a good approximation.


\url{https://en.wikipedia.org/wiki/Polynomial}
\url{https://en.m.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem}
\url{https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra}
\url{https://en.wikipedia.org/wiki/Vieta%27s_formulas}
\url{https://en.wikipedia.org/wiki/Root-finding_algorithms}
\url{https://mathworld.wolfram.com/MaclaurinSeries.html}