The terms of polynomials are the parts of the equation that are generally separated by “+” or “-” signs. So, each part of a polynomial in an equation is a term.

Polynomial 3x^2−9x+7 where x=3

3-9+7

\(3(3)^2−9(3)+7\)

\(3⋅9−9(3)+7\)

\(27−27+7\)

\(7\)

Before continuing our discussion of finite fields, we'd like to introduce the interesting subject of polynomial arithmetic. We are concerned with polynomials during a single variable x, and that we can distinguish three classes of polynomial arithmetic.

- Ordinary polynomial arithmetic, using the essential rules of the algebra.
- Polynomial arithmetic during which the arithmetic on the coefficients is performed modulo p; that's, the coefficients are in GF(p).
- Polynomial arithmetic during which the coefficients are in GF(p), and therefore the polynomials are defined modulo a polynomial m(x) whose highest power is a few integer n.

This section examines the primary two classes, and therefore the next section covers the last class.

A polynomial of degree n (integer n >= 0) is an expression of the shape

Where the ai are elements of some designated set of numbers S, called the coefficient set, and an != 0. We are saying that such polynomials are defined over the coefficient set S.

A zero-degree polynomial is named a continuing polynomial and is just a component of the set of coefficients. An nth-degree polynomial is claimed to be a polynomial if an = 1.

In the context of abstract algebra, we are usually not curious about evaluating a polynomial for a specific value of x [e.g., f(7)]. to stress now, the variable **x** is usually mentioned as indeterminate.

Polynomial arithmetic includes the operations of addition, subtraction, and multiplication. These operations are defined in a natural way as if the variable x was a component of S. Division is similarly defined but requires that S be a field. samples of fields include the important numbers, rational numbers, and Zp for p prime. Note that the set of all integers isn't a field and doesn't support polynomial division. Addition and subtraction are performed by adding or subtracting corresponding coefficients. Thus, if

In the last formula, we treat ai as zero for i > n and bi as zero for I> m. Note that the degree of the merchandise is adequate to the sum of the degrees of the 2 polynomials.

As an example, let f(x) = x3 + x2 + 2 and g(x) = x2 - x + 1, where S is that the set of integers. Then

f(x) + g(x) = x3 + 2x2 - x + 3

f(x) - g(x) = x3 + x + 1

f(x) * g(x) = x5 + 3x2 - 2x + 2

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