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



What is polynomial and what are the uses of it?


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.

Ordinary Polynomial Arithmetic

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


How to use this tool Ordinary Polynomial Arithmetic

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There are x^2x^1, and X where you will enter the value.

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