Polynomials

3 Key excerpts on "Polynomials"

• 

  eBook - ePub

  Elementary Algebra

  • Toby Wagner(Author)
  • 2021(Publication Date)
  • Chemeketa Press

    (Publisher)

  2 , two terms contain x, and two terms don’t contain any variable at all. The different powers of the variable must be taken into consideration when adding or subtracting. We’ll discuss why this is true and then return to this problem. This section will show you how to:

  ◆  identify Polynomials, terms, and degree ◆  identify and combine like terms ◆  simplify expressions containing parentheses ◆  add and subtract Polynomials

  A. Introduction to Polynomials

  What makes an expression a polynomial? Polynomials are algebraic expressions that do not contain variables in the denominators of fractions and that do have whole numbers as the exponents on the variable quantities. The table below shows some examples of expressions that are Polynomials.

  Expression	Why It’s a Polynomial
  3 x4	The exponent on the variable is a whole number.
  	There is a fraction, but no variable appears in the denominator. The exponents on the variables are whole numbers.
  5 x3 + 3 x2 – 2x + 1	The exponents on the variables are whole numbers.

  The table below offers some expressions that are not Polynomials:

  Expression	Why It’s Not a Polynomial
  	A variable appears in the denominator of the fraction
  4 x2 – 5x + x−3	A negative exponent appears on the variable in the third term.
  7 x	The exponent on the variable is not a whole number.

  The terms of a polynomial are the expressions being added together. When listing the terms of a polynomial, keep in mind that subtracting is the same as adding the opposite. For example, the expression 4x – 5 is mathematically equivalent to the expression 4x + (−5). A polynomial consists of only one term if it does not involve addition or subtraction.

  The coefficient refers to the constant factor of a term. For example, in the term 7x

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  eBook - ePub

  Algebra & Geometry

  An Introduction to University Mathematics

  • Mark Verus Lawson(Author)
  • 2016(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  quintic.

  For most of this book, we shall only deal with Polynomials in one variable, but in a few places we shall also need the notion of a polynomial in several variables. Let x1 ,…,xn be n variables. When n = 1 we usually take just x, when n = 2, we take x, y and when n = 3, we take x, y, z. These variables are assumed to commute with each other. It follows that a product of these variables can be written

  x 1

  x 1

  …

  x n

  r n

  . This is called a monomial of degree r1 + … + rn . If ri = 0 then

  x i

  r i

  = 1

  and is omitted. A polynomial in n variables with coefficients in F , denoted by

  F [

  x 1

  , … ,

  x n

  ]

  , is any sum of a finite number of distinct monomials each multiplied by an element of F . The degree of such a non-zero polynomial is the maximum of the degrees of the monomials that appear.

  Example 7.1.1. Polynomials in one variable of arbitrary degree are the subject of this chapter whereas the theory of Polynomials in two or three variables of degree 2 is described in Chapter 10 . The most general polynomial in two variables of degree 2 looks like this

  a

  x 2

  + b x y + c

  y 2

  + d x + e y + f .

  It consists of three monomials of degree 2: namely, x2 , xy and y2 ; two monomials of degree 1: namely, x and y; and a constant term arising from the monomial of degree 0. Polynomials of degree 1 give rise to linear equations which are discussed in Chapter 8

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• 

  eBook - ePub

  Galois' Theory of Algebraic Equations

  • Jean-Pierre Tignol(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  Chapter 5

  A Modern Approach to Polynomials

   

  5.1Definitions

  In modern terminology, a polynomial in one indeterminate with coefficients in a ring A can be defined as a map

  P: ℕ → A

  such that the set supp P = {n ∈ ℕ | Pn ≠ 0}, called the support of P, is finite. The addition of Polynomials is the usual addition of maps,

  (P + Q)

  n

  = Pn + Qn

  and the product is the convolution product

  Every element a ∈ A is identified with the polynomial a: ℕ → A that maps 0 to a and n to 0 for n ≠ 0. Denoting by

  X: ℕ → A

  the polynomial that maps 1 onto the unit element 1 ∈ A and the other integers to 0, it is then easily seen that every polynomial P can be uniquely written as

  Therefore, we henceforth write

  (as is usual!) for the polynomial that maps i ∈ ℕ to ai for i = 0, …, n and to 0 for i > n. Accordingly, the set of all Polynomials with coefficients in A (or Polynomials over A) is denoted A[X]. Straightforward calculations show that A[X] is a ring, which is commutative if and only if A is commutative.

  The ring of Polynomials in any number m of indeterminates over A can be similarly defined as the ring of maps from ℕ

  m

  to A with finite support, with the convolution product.

  Of course, the definition above looks somewhat artificial. The naive approach to Polynomials is to consider expressions like (5.1), where X is an undefined object, called an indeterminate, or a variable. While this terminology will be retained in the sequel, it should be observed that, without any other proper definition, to say something is an indeterminate or a variable is hardly a definition. Moreover, it fosters confusion between the polynomial

  P(X) = a0 + a1 X + ··· + an Xn

  and the associated polynomial function

  P(·): A → A

  which maps x ∈ A onto P(x) = a0 + a1 x + ··· + an xn . This same confusion has prompted the use of the term constant Polynomials for the elements of A, considered as Polynomials. While this confusion is not so serious when A is a field with infinitely many elements (see Corollary 5.16 below), it could be harmful when A is finite. For instance, if A = {a1 ,…, an } (with n ≥ 2), then the polynomial (X −a1 ) … (X −an ) is not the zero polynomial since the coefficient of Xn is 1, but its associated polynomial function maps every element of A

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