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# 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]
}
$$


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