Polynomials

11 Key excerpts on "Polynomials"

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  Practical MATLAB Basics for Engineers

  • Misza Kalechman(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  411 7 Polynomials and Calculus, a Numerical and Symbolic Approach I keep the subject constantly before me and wait till the first dawning open little by little into full light. Sir Isaac Newton 7.1 Introduction Polynomials are algebraic expressions consisting of the sum of one or more product terms. A function defined by a polynomial expression is referred to as a polynomial. In its simplest form, each of the product terms consists of a coefficient, and a variable of interest ( x ) raised to a nonnegative integer power. A polynomial {of one variable ( x )} can in general be expressed as f x a x a x a x a a x n n n n n ( ) 1 1 1 0 1 1 1 0 = ∑ where x is the variable, the a’ s ( a n , a n − 1 , a n − 2 , …, a 0 ) are the coefficients assuming that a n ≠ 0 , and the nonnegative integer n with the highest exponent defines the degree of the polynomial. Some of the most frequently used Polynomials are defined as follows: f ( x ) = 0, where f ( x ) has no degree f ( x ) = a 0 , where a 0 ≠ 0, and the degree of f ( x ) is zero f ( x ) = a 1 x + a 0 , where a 1 ≠ 0, the degree of f ( x ) is 1, and defines a linear relation f ( x ) = a 2 x 2 + a 1 x + a 0 , where a 2 ≠ 0, the degree of f ( x ) is 2, and defines a quadratic relation f ( x ) = a 3 x 3 + a 2 x 2 + a 1 x + a 0 , where a 3 ≠ 0, the degree of f ( x ) is 3, and defines a cubic relation Polynomials are extensively used in technology, engineering, and the sciences because they can best represent, or model a physical system. Polynomials are also easy to define and to evalu-ate because they involve the basic operations of additions, subtractions, and multiplications. In engineering and science, it is often required to determine the roots of the polynomial equation defined by f x a x m m m n ( ) 0 0 ∑ where in general the coefficients a m , for m = 0, 1, 2, 3, …, n , may be complex.

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  Introductory Experimental Mathematics & Key Mathematical Concepts

  • (Author)
  • 2014(Publication Date)
  • Orange Apple

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter- 5 Polynomial and Rational Function Polynomial In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x 2 − 4 x + 7 is a polynomial, but x 2 − 4/ x + 7 x 3/2 is not, because its second term involves division by the variable x (4/x) and because its third term contains an exponent that is not a whole number (3/2). The term 'polynomial' indicates a simplified algebraic form such that all Polynomials are similarly simple in complexity (cf. polynomial time). Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, Polynomials are used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry. Overview A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) which may be multiplied by a finite number of variables (usually represented by letters). Each variable may have an exponent that is a non-negative integer, i.e., a natural number. The exponent on a variable in a term is called the degree of that variable in that term, the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any one term.

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  Advanced Linear Algebra

  • Bruce Cooperstein(Author)
  • 2015(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  3 Polynomials CONTENTS 3.1 The Algebra of Polynomials ..................................... 88 3.2 Roots of Polynomials ............................................ 99 In this chapter, we build on high school algebra and develop the algebraic theory of Polynomials. In section one we show that under the usual operations of addition and multiplication the collection of all Polynomials with coefficients in a field F is a commutative algebra with identity. We define the concepts of greatest common divisor (gcd) and least common multiple (lcm) of two Polynomials and make use of the division algorithm (division with remainders) to establish the existence and uniqueness of the gcd and lcm. In section two we prove some general results about roots of Polynomials and then specialize to Polynomials with coefficients in the fields R and C . 87 88 Advanced Linear Algebra 3.1 The Algebra of Polynomials What You Need to Know Elementary properties of Polynomials, such as how to add and multiply poly-nomials and how to compute the quotient and remainder when one polynomial is divided by another. We begin by recalling the definition of a polynomial in a variable x and intro-duce some notation and terminology which will facilitate the discussion. Definition 3.1 Let F be a field. By a polynomial with coefficients in F , we mean an expression of the form a m x m + a m − 1 x m − 1 + · · · + a 1 x + a 0 , where a i ∈ F and x is an abstract symbol called an indeterminate or variable . The scalars a i are the coefficients of the polynomial f ( x ) . The zero polynomial is the polynomial all of whose coefficients are zero. We denote this by 0. Suppose f ( x ) negationslash = 0 . The largest natural number k such that the coefficient a k is not zero is called the degree of f ( x ) and the term a k x k is called the leading term . If the coefficient of the leading term is 1 we say the polynomial f ( x ) is monic .

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  Practical Algebra

  A Self-Teaching Guide

  • Bobson Wong, Larisa Bukalov, Steve Slavin(Authors)
  • 2022(Publication Date)
  • Jossey-Bass

    (Publisher)

  2 x 2 + 0 x + 4 + 6 x 2 + 5 x 8 x 2 + 5 x + 4 233 234 PRACTICAL ALGEBRA When talking about expressions with variables raised to a power greater than 1, we need some vocabulary: • A polynomial is the sum or difference of one or more terms. Examples of polyno-mials include a monomial (which has one term), a binomial (which has two terms), and a trinomial (which has three terms). • The degree of a term is the sum of the exponents of the variables in the term. The degree of a polynomial is the largest of all of these sums. Most of the time, you’ll only encounter one variable, so in those cases you can think of the degree of a polynomial as the highest exponent of the variable. For example • x 5 + 2 x 2 + 4 x has degree 5 (The polynomial has only one variable, whose highest exponent is 5.) • x 2 + xyz + z 2 has degree 3 (The terms have degrees 2, 1 + 1 + 1 = 3, and 2, respectively. The highest of these degrees is 3.) • We typically write Polynomials in standard form , which means that terms decrease in degree from left to right and all like terms are combined. Table 8.1 shows examples and non-examples of Polynomials. Table 8.1 Polynomials. Example Reason Non-Example Reason 5 a 2 − 3 a − 7 This polynomial is a trinomial—the sum of three terms. q − 5 3 q This expression cannot be rewritten as a product of numbers and variables. There is a variable in the denominator. 6 A polynomial may consist of only one term—in this case, the number 6. n 2 + 7 n + 3 + 1 n + 6 There is a variable in the denominator of 1 n + 6 . x 7 This can be rewritten as 1 7 x , which is a monomial—the product of the number 1 7 and the variable x . 7 x There is a variable in the denominator. Watch Out! As said in Chapter 1, we read x 2 as “ x -squared,” “ x to the second power,” or “ x to the second,” not “ x -two.” Similarly, we read x 3 as “ x -cubed,” “ x to the third power,” or “ x to the third,” not “ x -three.”

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  Topics In Polynomials: Extremal Problems, Inequalities, Zeros

  Extremal Problems, Inequalities, Zeros

  • Gradimir V Milovanovic, Themistocles M Rassias, D S Mitrinovic(Authors)
  • 1994(Publication Date)
  • World Scientific

    (Publisher)

  C H A P T E R 1 General Concepts of Algebraic Polynomials 1.1. P O L Y N O M I A L S A N D E Q U A T I O N S 1.1.1. Preliminaries Let F be a field. The ring of Polynomials in x over F, denoted by F[i], is defined to be the set of all formal expressions P(x) = aa + atx + + a n x n , where the a;, the coefficients of the polynomial P in x, are elements in F. If a n ^ 0, then the degree of P, denoted by dg P, is n. Therefore, the degree of a polynomial P is the highest power of x that occurs in the expression for P(x) with a non-zero coefficient. In F[i] we define equality, sum, and product of two Polynomials as follows. Given two Polynomials P(x) = a 0 + a 1 x + --- +a n x n and Q(x) = b 0 + b i x + ---+b m x m , then the Polynomials are equal if, and only if, a; = 6; for all i > 0, that is when their corresponding coefficients are equal. We define the sum of P and Q by {P + Q)(x) = P(x) + Q(x) = c 0 + ax + • • • + c r x r , l 2 GENERAL CONCEPTS OF Polynomials where for each t, C; = a; + (>;, and the product of P and Q by (PQ)(x) - P(x)<30r) = co + cia; + • + c,*', where c,- = a,6 0 + <*i-i&i + h ai&i_i + a 0 b„ for every i . Then, one can show that F[x] is a commutative ring with unit. It is easy to prove that if P(x) and Q(x) are non-zero elements of V[x], then dg(PQ) = dgP + dgt?. Also, if P{x),Q{x) e F[x] and P(x) + Q(x) ± 0, then dgtP + g j ^ m a x I d g ^ d g Q } . We do not assign a degree to 0{x) = Ox + 0 x n _ 1 + • • • + Ox + 0. The Polynomials of degree 0 are called the constants (these are the elements of the field F). Let k ,... , k m be non-negative integers and fc = ( * i , . . . , A m ) , = + + x k = »f» • * * » .

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  Abstract Algebra

  A Gentle Introduction

  • Gary L. Mullen, James A. Sellers(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 7 Polynomials 7.1 Basics ............................................................. 111 7.2 Unique factorization ............................................. 115 7.3 Polynomials over the real and complex numbers ................ 117 7.4 Root formulas .................................................... 118 In this chapter we discuss Polynomials, mathematical objects with which we are very familiar from our studies in algebra and calculus. Polynomials occur in almost every mathematical setting. Even though the reader is likely to be familiar with Polynomials, we begin by defining them rather carefully. We will consider Polynomials defined over both rings and fields. 7.1 Basics Assume that R is a commutative ring. The ring R might well be an integral domain or a field, but for now, we just assume that R is a commutative ring. A polynomial with coefficients in R is an expression of the form n p ( x ) = a n x + a n − 1 x n − 1 + + a 1 x + a 0 · · · where the coefficients a n , . . . , a 0 ∈ R , x is an indeterminate or a variable that can take on values from the ring R , and n ≥ 0 is a non-negative integer. A polynomial is monic if the coefficient of the highest power of x is 1, i.e., in our notation above, if a n = 1. A polynomial p ( x ) has degree n if the coefficient a n of the highest power of x is non-zero. Thus every polynomial except the zero polynomial (whose coefficients are all 0) has a degree that is at least 0. We define the degree of the zero polynomial to be − 1. Let p ( x ) = a n x n + + a 0 · · · q ( x ) = b m x m + + b 0 (7.1) · · · be two Polynomials with coefficients in the real numbers where we may assume 111 112 Abstract Algebra: A Gentle Introduction that m ≤ n . Then as Polynomials, p ( x ) = q ( x ) if the coefficients of each power of x are equal, i.e., if a i = b i , i = 0 , . . . , m (and a m +1 = = a n = 0). · · · Recall that two functions f and g are equal as functions on a ring R if f ( a ) = g ( a ) for all a ∈ R .

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  Introduction to Mathematics with Maple

  • P Adams, K Smith;R V??born??;;(Authors)
  • 2004(Publication Date)
  • WSPC

    (Publisher)

  Chapter 6 Polynomials Polynomial functions have always been important, if for nothing else than because, in the past, they were the only functions which could be readily evaluated. In this chapter we define Polynomials as algebraic entities rather than functions, establish the long divi- sion algorithm in an abstract setting, we also look briefly at zeros of Polynomials and prove the Taylor Theorem for Polynomials in a generality which cannot be obtained by using methods of calculus. 6.1 Polynomial functions If M is a ring and ao, al, a2, . . . , an E M then a function of the form is called a polynomial, or sometimes more explicitly, a polynomial with coefficients in M . Obviously, one can add any number of zero coefficients, or rewrite Equation (6.1) in ascending order of powers of z without changing the polynomial. The domain of definition of the polynomial is naturally M , but the definition of A(x) makes sense for any x in a ring which contains M . This natural extension of the domain of definition is often understood without explicitly saying so. If A and B are two Polynomials then the Polynomials A + B, -A and AB are defined in the obvious way as A + B : x H A(x) + B(x) -A : x -A(x) AB : z H A(x)B(x) The coefficients of A+B are obvious; they are the sums of the corresponding coefficients of A and B. The zero polynomial function is the zero function, 167 168 Introduction to Mathematics with Maple that is 10 : x H 0. Similarly, the coefficients of -A have opposite signs to the coefficients of A. The coefficients of AB are obtained by multiplying through, collecting terms with the same power of x and sorting them in descending (or ascending) powers of x. If and P = AB, then There is a clear pattern to the formulae + an-ibm (6.2). In order to subsume them in a compact formula we set a k = 0 for k > n and bk = 0 for k > rn. Then we can rewrite Equations (6.2) as k pk = ak-jbj j=O for' k = 1,2,.

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  Mathematical Objects in C++

  Computational Tools in A Unified Object-Oriented Approach

  • Yair Shapira(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 12 Polynomials A real polynomial is a function p : R → R defined by p ( x ) ≡ a 0 + a 1 x + a 2 x 2 + · · · + a n x n = n i =0 a i x i , where n is a nonnegative integer called the degree of the polynomial, and a 0 , a 1 , a 2 . . . , a n are given real numbers called the coefficients. Thus, to define a concrete polynomial it is sufficient to specify its coefficients a 0 , a 1 , a 2 . . . , a n . Thus, the polynomial is equivalent to the ( n +1)-dimensional vector ( a 0 , a 1 , a 2 . . . , a n ) . A complex polynomial is different from a real polynomial in that the coeffi-cients a 0 , a 1 , a 2 . . . , a n , as well as the variable x , can be not only real but also complex. This makes the polynomial a complex function p : C → C . 12.1 Adding Polynomials The interpretation of Polynomials as ( n + 1)-dimensional vectors is useful in some arithmetic operations. For example, if another polynomial of degree m ≤ n q ( x ) ≡ m i =0 b i x i is also given, then the sum of p and q is defined by ( p + q )( x ) ≡ p ( x ) + q ( x ) = n i =0 ( a i + b i ) x i , where, if m < n , one also needs to define the fictitious zero coefficients b m +1 = b m +2 = · · · = b n ≡ 0 . (Without loss of generality, one may assume that m ≤ n ; otherwise, the roles of p and q may interchange.) 203 204 CHAPTER 12. Polynomials In other words, the vector of coefficients associated with the sum p + q is the sum of the individual vectors associated with p and q (extended by leading zero components if necessary, so that they are both ( n + 1)-dimensional). Thus, the interpretation of Polynomials as vectors of coefficients helps us to add them by just adding their vectors. 12.2 Multiplying a Polynomial by a Scalar The representation of the polynomial as a vector also helps to multiply it by a given scalar a : ( ap )( x ) ≡ a · p ( x ) = a n i =0 a i x i = n i =0 ( aa i ) x i .

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  Numbers, Groups and Codes

  • J. F. Humphreys, M. Y. Prest(Authors)
  • 2004(Publication Date)
  • Cambridge University Press

    (Publisher)

  We wish to regard these three expressions (and all others we can get from them by adding terms with 0 coefficient and by rearranging terms) as being ‘the same’ polynomial. In other words, we regard two polynomial expressions as being equivalent if we can get from one to the other by rearranging terms and 255 256 Polynomials adding or deleting terms with 0 coefficient. It is normal to say and write that such expressions are ‘equal’ rather than ‘equivalent’: so we write, for example, − 1 + 2 x + x 3 = x 3 + 2 x − 1 . A typical polynomial can, therefore, be written in the form a 0 + a 1 x + · · · + a i x i + · · · If we want to make this look more uniform we may write a 0 x 0 + a 1 x 1 + · · · + a i x i + · · · We say that a i x i is a term of the polynomial and that a i is the coefficient of x i . We do require that a polynomial should only have finitely many non-zero terms, that is, a i = 0 for all but a finite number of values of i . We say that the power x i appears in the polynomial if a i = 0. So x 3 appears in x 3 + 0 x 2 + 2 x − 1 but x 2 does not. We use notation such as f ( x ) , g ( x ) , r ( x ), etc. for Polynomials but sometimes we drop the ‘( x )’, writing f , g , r etc. We also use the same notation for the functions defined by Polynomials. Summation notation gives a compact way of writing Polynomials: in this notation a typical polynomial f ( x ) has the form f ( x ) = ∑ n i = 0 a i x i ; the other way of writing this is a 0 + a 1 x + · · · + a n x n (where we have replaced a 0 x 0 which equals a 0 · 1 by a 0 , and a 1 x 1 is written more simply as a 1 x ). If a n = 0 , in other words if x n is the highest power of x which appears in the polynomial, then we call a n x n and a n the leading term and leading coefficient respectively and we say that that the degree of f ( x ) is n and write deg( f ( x )) = n . For example the degree of x 3 + 2 x − 1 is 3. The zero polynomial is a special case since it has no non-zero coefficients.

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  Computer Algebra and Symbolic Computation

  Mathematical Methods

  • Joel S. Cohen(Author)
  • 2003(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  6 Multivariate Polynomials Most symbolic computation involves mathematical expressions that con-tain more than one symbol or generalized variable. In order to manipulate these expressions, we must extend the concepts and algorithms described in Chapter 4 to Polynomials with several variables. This generalization is the subject of this chapter. It includes a description of coefficient domains and a recursive view of multivariate Polynomials (Section 6.1), versions of polynomial division and expansion (Section 6.2), greatest common divi-sor algorithms (Section 6.3), and the extended Euclidean algorithm (Sec-tion 6.3). 6.1 Multivariate Polynomials and Integral Domains A multivariate polynomial u in the set of distinct symbols {#i, #2,..., x p } is a finite sum with (one or more) monomial terms of the form where the coefficient c is in a coefficient domain K and the exponents rij are non-negative integers. (The axioms for K (which may not be a field) are given in Definition 6.2 below.) The notation K[xi,X2,... ,x p ] represents the set of Polynomials in the symbols #i, #2, • • • > Xp with coefficients in K. For example, Z[#,2/] represents all Polynomials in x and y with coefficients that are integers. A particularly important instance when K is not a field has to do with the recursive representation of multivariate Polynomials. For example, let 201 202 6. Multivariate Polynomials Q[x, y] be the Polynomials in x and y with rational number coefficients. By collecting coefficients of powers of x, a polynomial u(x,y) is represented as u(x, y) = Um(y)x m + Um-^x™' 1 + • • • + u 0 {y), where the coefficients Ui(y) are in Q[y]. In this sense, Q[X,Î/] is equivalent to K.[x] where the coefficient domain K = Q[y]. In general, we can view K[xi, a?2,..., #p] recursively as K[a?i,a?2,...,Xp] = Ki[a?i], (6.1) where K¿ = K¿ + i[x¿+i], 1 < i < p -1, and K p — K.

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  Intermediate Algebra

  A Guided Approach

  • Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  316 CHAPTER 5 Polynomials, Polynomial Functions, and Equations 132. Geometry Write a polynomial that represents the area of the rectangular field. ( x – 2) ft ( x + 4) ft 133. Geometry Write a poly-nomial that represents the area of the triangle. 134. Geometry Write a polynomial that represents the volume of the cube. WRITING ABOUT MATH 135. Explain how to use the distributive property or “FOIL method” to multiply two binomials. 136. Explain how to multiply two trinomials. SOMETHING TO THINK ABOUT 137. The numbers 0.35 3 10 7 and 1.96 3 10 7 both involve the same power of 10. Find their sum. 138. Without converting to standard notation, find the sum: 1.435 3 10 8 1 2.11 3 10 7 . ( Hint : The first number in the previous exercise is not in scientific notation.) ( b + 5) in. ( b – 2) in. ( x + 2) ft Section 2 3 Objectives Vocabulary Find the prime-factored form of a natural number. Find the greatest common factor (GCF) of two or more monomials. Factor the greatest common factor (GCF) from a polynomial. Factor a polynomial with four or six terms by grouping. Solve a formula for a specified variable by factoring. 1 The Greatest Common Factor and Factoring by Grouping factoring prime-factored form greatest common factor (GCF) prime polynomial 4 5 Simplify each expression by using the distributive property to remove parentheses . 1. 4 1 a 1 2 2 2. 2 5 1 b 2 5 2 3. a 1 a 1 5 2 4. 2 b 1 b 2 3 2 5. 6 a 1 a 1 2 b 2 6. 2 2 p 2 1 3 p 1 5 2 Getting Ready 5.4 Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.

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