Polynomial

https://en.wikipedia.org/wiki/Polynomial

A polynomial is an expression consisting of variables (indeterminates) and coefficients (constants) that involves only the operations of addition, subtraction, multiplication and non-negative integer exponents of variables.

For example, a polynomial with 1 variable (x); polynomial with 3 variables (x,y,z):

$$x^2 − 4x + 7 \\ x^3 + 2xyz^2 − yz + 1 \\$$

Polynomials appear in many areas: polynomial equations, polynomial functions, elementary word problem, complex scientific problems; they are used in calculus,numerical analysis, algebra, algebraic geometry.

A polynomial in a single indeterminate $x$ can always be written in the form:

$a_{n} x^{n} + a_{n-1} x^{n-1} + \dotsb + a_{2} x^{2} + a_{1} x+a_{0}$

where $a_{0},\ldots ,a_{n}$ are constants and $x$ is the indeterminate.

Expressed concisely using summation:

${\displaystyle \sum _{k=0}^{n}a_{k}x^{k}}$

That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a number, called the coefficient of the term, and a finite number of indeterminates, raised to nonnegative integer powers.

Quadratic equation

$$\displaystyle{ \Big(\sum_{i=1}^n a_i\Big)^2 = \sum_{i=1}^n (a_i)^2 + 2\Big[\sum_{j=i}^{n-1} \Big( a_j \sum_{k=i+1}^n a_{k} \Big) \Big] }$$

$$\displaystyle{ \Big(\sum_{i=1}^n a_i\Big)^2 = (a_1 + a_2 + \dotsb + a_n)^2 = q + 2p }$$

$$\displaystyle{ q = \sum_{i=1}^n a_i^2 = a_1^2 + a_2^2 + \dotsb + a_n^2 }$$

$$\displaystyle{ p = a_1(a_2 + \dotsb + a_n) + a_2(a_3 + \dotsb + a_n) + \dotsb + a_{n-2}(a_{n-1} + a_n) + a_{n-1}(a_n) }$$

$$\displaystyle{ p = a_1\Big(\sum_{i=2}^n a_i\Big) + a_2\Big(\sum_{i=3}^n a_i\Big) + \dotsb + a_{n-2}\Big(\sum_{i=(n-1)}^n a_i\Big) + a_{n-1}\Big(\sum_{i=n}^n a_i\Big) }$$

$$\displaystyle{ p = \sum_{i=1}^{n-1} \Big[ a_i \Big(\sum a_{i+1}\Big) \Big] }$$