Mathematics
Operations with Polynomials
Operations with polynomials involve performing arithmetic operations such as addition, subtraction, multiplication, and division on polynomial expressions. When adding or subtracting polynomials, like terms are combined. Multiplying polynomials involves distributing each term of one polynomial across the other and then combining like terms. Division of polynomials is the process of finding the quotient and remainder when one polynomial is divided by another.
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12 Key excerpts on "Operations with Polynomials"
eBook - PDFPractical Algebra
A Self-Teaching Guide
- Bobson Wong, Larisa Bukalov, Steve Slavin(Authors)
- 2022(Publication Date)
- Jossey-Bass(Publisher)
8 Operations with Polynomials So far, we’ve focused on linear expressions. In this chapter, we’ll discuss operations (addition, subtraction, multiplication, and division) with non-linear expressions, whose variables have exponents that are greater than 1. 8.1 Adding and Subtracting Polynomials Let’s start by looking at an addition problem: 204 + 650. What’s wrong with the fol-lowing work? 24 + 65 89 It ignores the importance of 0 as a placeholder for the tens in 204 and for the zeros in 650. If we rewrite 204 and 650 in expanded form (in other words, as powers of 10), we get the following: 2 ( 10 2 ) + 0 ( 10 1 ) + 4 ( 10 0 ) + 6 ( 10 2 ) + 5 ( 10 1 ) + 0 ( 10 0 ) 8 ( 10 2 ) + 5 ( 10 1 ) + 4 ( 10 0 ) Writing the numbers in this way shows that we need to “line up” the digits with the same place value. Adding 4(10 0 ) and 5(10 1 ) to get 9(10 0 ) is incorrect. The number system that we use, which was developed in India and brought to Europe through the work of Middle Eastern mathematicians such as al-Khw¯ arizm¯ ı and al-Kindi, is based on powers of 10. Since the bases in all of the powers here are 10, we call this a base-10 system. But it would apply just as well if we used a different base. (In fact, we use different bases all the time. Our division of hours into minutes and seconds is a product of the base-60 number system used by Mesopotamians thousands of years ago!) If we replaced the numerical base with a variable like x , then we see that to add these expressions, we need to “line up” the like terms and add up their coefficients. 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).
eBook - PDF- Dale Ewen(Author)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
◆ Simplify algebraic expressions by removing parentheses and combining like terms. ◆ Add and subtract polynomials. ◆ Multiply monomials. ◆ Multiply polynomials. ◆ Divide a monomial and a polynomial by a monomial. ◆ Divide a polynomial by a polynomial. 200 CHAPTER 5 ◆ An Introduction to Algebra In arithmetic, we perform mathematical operations with specific numbers. In algebra, we perform these same basic mathematical operations with numbers and variables —letters that represent unknown quantities. Algebra allows us to express and solve general as well as specific problems that cannot be solved using only arithmetic. As a result, employers in technical and scientific areas require a certain level of skill and knowledge of algebra. Your problem-solving skills will increase significantly as your algebra skills increase. To begin our study of algebra, some basic mathematical principles that you will apply are listed below. Most of them you probably already know; the rest will be discussed. Note that “ Þ ” means “is not equal to.” Basic Mathematical Principles 1. a 1 b 5 b 1 a (Commutative Property for Addition) 2. ab 5 ba (Commutative Property for Multiplication) 3. ( a 1 b ) 1 c 5 a 1 ( b 1 c ) (Associative Property for Addition) 4. ( ab ) c 5 a ( bc ) (Associative Property for Multiplication) 5. a ( b 1 c ) 5 ab 1 ac , or ( b 1 c ) a 5 ba 1 ca (Distributive Property) 6. a 1 0 5 a 7. a # 0 5 0 8. a 1 ( 2 a ) 5 0 (Additive Inverse) 9. a # 1 5 a 10. a # 1 a 5 1 ( a Þ 0) (Multiplicative Inverse) In mathematics, letters are often used to represent numbers. Thus, it is necessary to know how to indicate arithmetic operations and carry them out using letters. Addition: x 1 y means add x and y . Subtraction: x 2 y means subtract y from x or add the negative of y to x ; that is, x 1 ( 2 y ). Multiplication: xy or x # y or ( x )( y ) or ( x ) y or x ( y ) means multiply x by y . Division: x 4 y or x y means divide x by y , or find a number z such that zy 5 x .
eBook - PDF- 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,.
eBook - PDF- Alberto D. Yazon(Author)
- 2019(Publication Date)
- Arcler Press(Publisher)
Algebra and Basic Math 2 CONTENTS 2.1. Fundamentals Of Algebra ................................................................. 38 2.2. Operations On Monomials and Polynomials ..................................... 40 2.3. Linear Equations In One Variable ...................................................... 46 2.4. Problems To Solve ............................................................................. 58 References ............................................................................................... 64 Fundamentals of Advanced Mathematics 38 Mathematics is a vast subject, which includes arithmetic, calculations, statistics, and various other branches. One of the significant fields that make up the great subject of mathematics is ‘Algebra.’ Algebra is the basis for all the topics that come up in the advanced form of mathematics like calculus, relations, and functions and other important topics. Hence it becomes important for the readers and learners to revise and view the basic concepts of algebra before moving forward. 2.1. FUNDAMENTALS OF ALGEBRA There is a very significant part of mathematics which is made up by numbers and some general rules of arithmetic and that part is commonly known as ‘algebra.’ Algebra takes the methods and learnings of arithmetic in such a way that it becomes easy to imply the rules related with the calculation of numbers and make use of these rules to work with some symbols too other than the num-bers. The adoption of algebra provides easy access to several other branches of mathematics, rather than an abrupt makeover into new fields, with the use of previously attained knowledge of the use of basic arithmetical operations. The way of writing quantities in some common ways instead of a particular set of arithmetic terms is widely known.
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
48 Module 2 • The four basic algebraic operations Adding and subtracting in algebra So far, we have added and subtracted numbers. We know that in algebra we use letters as placeholders. We have also learnt in Module 1 that a term has a coefficient, a coefficient has a sign, a term has a base with or without an exponent and the base can be raised to a power. –3 a 2 term sign coefficient to the power exponent or index base Pre-knowledge Definitions Algebraic expression An algebraic expression is one or more algebraic terms in a phrase. It can include variables , constants and operating symbols, such as plus and minus signs. It is only a phrase, not a whole sentence, so it does not include an equal sign . Example: 3 x 2 – 2 y + 7 xy + y 2 – 5 Term Terms are elements that are separated by a plus or a minus sign only . The example above has five terms: 3 x 2 , –2 y , 7 xy , y 2 and –5. The sign in front of a term belongs to that term. Terms could consist of variables and coefficients, or constants. Variable A variable is a letter or symbol that represents an unknown value. In algebraic expressions letters represent variables. These letters are actually numbers in disguise. In the expression 3 x 2 – 2 y + 7 xy + y 2 – 5 the variables are x and y . We call these letters var iables because the numbers they represent can vary : we can substitute one or more numbers for the letters in the expression. Coefficient A coefficient is the number, together with its sign, which is multiplied by the variable in an algebraic expression. In 3 x 2 – 2 y + 7 xy + y 2 – 5 the coefficient of the first term is 3 , the coefficient of the second term is – 2 , the coefficient of the third term is 7 and the coefficient of the fourth term is 1 . If a term consists of only variables, as in y 2 , its coefficient is 1. Constant A constant is a number that cannot change its value. Constants are the terms in the algebraic expression that contain only numbers: they are the terms without variables.
eBook - PDF- J. F. Humphreys, M. Y. Prest(Authors)
- 2004(Publication Date)
- Cambridge University Press(Publisher)
We emphasise that the facts we have needed to use in our discussion are the basic properties (such as commutativity and distributivity) of the algebraic operations on R : it is these which underpin the corresponding properties of R [ x ]. For this reason, we could equally have considered C [ x ], the set of polynomials with complex coefficients of a complex variable x , in place of R [ x ]. The definition of addition and multiplication of polynomials would be as above and we would have exactly the same basic algebraic properties as for R [ x ]. However, one major difference is that every non-constant polynomial, f , in C [ x ] has a zero: that is, there is a complex number α such that f ( α ) = 0. This process, changing coefficients, does not end here. Given any prime number p , we can consider (as in Chapter 1) the set, Z p , of congruence classes modulo p . Again, this set is a ring and so we can consider Z p [ x ], the set of polynomials over Z p (that is, with coefficients in Z p ). Every-thing (definitions and basic algebraic properties) is as before. We do, of course, when adding and multiplying such polynomials, have to calculate the coeffi-cients of the sum and the product using arithmetic modulo our prime p . Thus if p = 2 , ( x 2 + x + 1) + ( x 2 + 1) = x and when p = 3 , ( x + 2) 2 = x 2 + x + 1. Similarly, when evaluating such polynomials, we have to calculate modulo p . You might wonder why we do this only for prime numbers p : surely in almost everything we have said so far we could replace the prime p by any integer n ≥ 2. That is correct and it is really only when we come to division (in the next section) that we do need p to be prime. For remember that if n is not prime then Z n is not a field and this means that we cannot always divide by non-zero elements. For the next section we certainly need to be in a context where we can always divide by non-zero coefficients. There is an important feature of polynomials with coefficients in Z p .
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
Don’t try to verify the equivalence of expressions by substituting 0, 1, or 2 for the variable, as using these numbers will occasionally give you false results. It is not good practice to trust your intuition or instincts in every new situation in algebra. If you have any doubt about the generalizations you are making, test them by replacing variables with numbers and simplifying. © Lajos Repasi / iStockPhoto The study skills for this chapter cover the way you approach new situations in mathematics. The first study skill is a point of view you hold about your natural instincts for what does and doesn’t work in mathematics. The second study skill gives you a way of testing your instincts. 325 5.1 Learning Objectives In this section, we will learn how to: 1. Give the degree of a polynomial. 2. Add and subtract polynomials. 3. Evaluate a polynomial for a given value of its variable. Sums and Differences of Polynomials Introduction We begin this section with the definition around which polynomials are defined. Once we have listed all the terminology associated with polynomials, we will show how the distributive property is used to find sums and differences of polynomials. Polynomials in General A term , or monomial , is a constant or the product of a constant and one or more variables raised to whole-number exponents. term, or monomial DEFINITION The following are monomials, or terms: − 16 3 x 2 − 2 _ 5 a 3 b 2 c xy 2 z The numerical part of each monomial is called the numerical coefficient , or just coefficient . For the preceding terms, the coefficients are − 16, 3, − 2 _ 5 , and 1. Notice that the coefficient for xy 2 z is understood to be 1. A term consisting only of a constant is called a constant term . For example, in the preceding terms, − 16 is a constant term. A constant term has degree 0. For a term containing a single variable, the degree is the exponent on the variable.
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
2 MODULE The four basic algebraic operations 2.1 The scientific pocket calculator On completion of this topic, you should be able to: 2.1.1 Do calculations including the four basic operations, extraction of roots and involution (inverse) with the aid of the scientific calculator 2.1.2 Use the memory keys in calculations 2.1.3 Use special keys in calculations, for example , , or as shown on most modern calculators, , , and so on. 2.2 The four basic operations On completion of this topic, you should be able to: 2.2.1 Add and subtract similar exponential terms 2.2.2 Multiply a monomial or a binomial expression by a monomial, binomial or trinomial expression, for example: i) 3 a (2 a + 5 a + 6) ii) (2 a + 3 b )(3 a + 6) iii) (2 a + 3)(3 a 2 + 6 a + 7) 2.2.3 Apply long division of a polynomial by a denominator not exceeding a binomial of the first degree, for example: (3 a 3 + 5 a 2 + 6 a + 7) ÷ (3 a + 2) 40 Module 2 • The four basic algebraic operations 2.1 The scientific pocket calculator Pre-knowledge • The order of operations: BODMAS or BIDMAS • 1 2 means one divided by two and 3 4 means three divided by four . If a sum contains more than one mathematical operational sign (+; –; ×; ÷), we will get a different answer if we do not all follow the same operational priority . The order in which operations are performed is BODMAS or BIDMAS . BODMAS • B rackets • O f • D ivision If there are no other operational signs between ÷ and ×, • M ultiplication 14243 we do the operations as they appear from left to right. • A ddition and s ubtraction as they appear from left to right after priority has been given to the previous four operations. BIDMAS • B rackets • I ndices • D ivision If there is no other operational signs between ÷ and ×, • M ultiplication 14243 we do the operations as they appear from left to right. • A ddition and s ubtraction as they appear from left to right after the priority has been given to the previous four operations.
eBook - PDF- Charles P. McKeague, Kate Duffy Pawlik(Authors)
- 2014(Publication Date)
- XYZ Textbooks(Publisher)
Various polynomials classified by number of terms and by degree: 3x 5 + 2x 3 + 1 A trinomial of degree 5 2x + 1 A binomial of degree 1 3x 2 + 2x + 1 A trinomial of degree 2 3x 5 A monomial of degree 5 −9 A monomial of degree 0 There are no new rules for adding one or more polynomials. We rely only on our previous knowledge. Here are some examples. Add (2x 2 − 5x + 3) + (4x 2 + 7x − 8). Solution We use the commutative and associative properties to group similar terms together and then apply the distributive property to add (2x 2 − 5x + 3) + (4x 2 + 7x − 8) = (2x 2 + 4x 2 ) + (−5x + 7x) + (3 − 8) Commutative and associative properties = (2 + 4)x 2 + (−5 + 7)x + (3 − 8) Distributive property = 6x 2 + 2x − 5 Addition The results here indicate that to add two polynomials, we add coefficients of simi- lar terms. Polynomial A polynomial is a finite sum of monomials (terms). Degree The degree of a polynomial in one variable is the highest power to which the variable is raised. VIDEO EXAMPLES SECTION 10.4 Example 1 632 Chapter 10 Exponents and Polynomials Add x 2 + 3x + 2x + 6. Solution The only similar terms here are the two middle terms. We combine them as usual to get x 2 + 3x + 2x + 6 = x 2 + 5x + 6 You will recall from Chapter 1 the definition of subtraction: a − b = a + (−b ). To subtract one expression from another, we simply add its opposite. The letters a and b in the definition can each represent polynomials. The opposite of a polynomial is the opposite of each of its terms. When you subtract one polynomial from another you subtract each of its terms. Subtract (3x 2 + x + 4) − (x 2 + 2x + 3). Solution To subtract x 2 + 2x + 3, we change the sign of each of its terms and add. If you are having trouble remembering why we do this, remember that we can think of −(x 2 + 2x + 3) as −1(x 2 + 2x + 3).
eBook - PDFAbstract Algebra
A Gentle Introduction
- Gary L. Mullen, James A. Sellers(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
· · · · · · In this notation, a 0 + b 0 is the sum of the elements a 0 and b 0 in the ring R , so the addition operation used is the additive operation for the ring R . Subtraction of polynomials works in a similar fashion. We note that even though we are working over a commutative ring R , the way we add and multiply polynomials over the ring is the same as we do it over the real numbers. For multiplication, we have the product p ( x ) q ( x ), which is given by summationdisplay k n + m a 0 b 0 + ( a 0 b 1 + a 1 b 0 ) x + + a i b j x + + a n b m x . · · · · · · i + j = k We note that if the ring R is an integral domain, then the degree of the product of two polynomials is the sum of the degrees of the individual poly-nomials. � Polynomials 113 The reader should check that if R is a commutative ring, then so is R [ x ], the set of all polynomials whose coefficients lie in the ring R . In fact, if the ring R is an integral domain, then R [ x ] is also an integral domain. We also mention that the ring R itself can be viewed as a subset of the polynomial ring R [ x ] by viewing an element a ∈ R as a polynomial of degree 0 if a = � 0, and of degree − 1 if a = 0. The reader will recall the Division Algorithm for integers from Theorem 1.2. It turns out that there is a polynomial version of the Division Algorithm that holds for polynomials over any field. In fact, the proof of the polynomial version is very similar to that given for the integer version, so we will omit this proof and simply state the algorithm: Theorem 7.1 (Division Algorithm) Let F be a field. Let f, g be two polyno-mials over F where f = 0 . Then there are unique polynomials q and r with the degree of r less than the degree of f so that g = fq + r . The polynomials q and r are often called the quotient and remainder polynomials. We now provide an example to illustrate the Division Algorithm over the field Z 2 of integers modulo 2.
eBook - PDFTopics 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» • * * » .
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
62 Introduction to Algebra ◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to • Define common algebraic terms: variable, expression, term, polynomial, and so forth. • Identify and define an equation. • Separate a term into variables and constants. • Simplify an expression by removing symbols of grouping. • Add and subtract polynomials. • Use the laws of exponents for multiplication, division, and raising to a power. • Multiply monomials, binomials, and multinomials. • Raise a multinomial to a power. • Divide a polynomial by a monomial or by a polynomial. ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ In Chapter 1 we showed how to raise a number to a power. For example, , or 3 raised to the power 2, means In a similar way, , or x raised to the power 2, means where x can stand for any number, not just 3. Going further, we can represent the exponent by a symbol, say n. Thus n factors While was an arithmetic expression, is an algebraic expression. We can think of algebra as a generalization of arithmetic. Some knowledge of algebra is essential in technical work. Suppose, for example, you see in a handbook that the power P delivered to a resistor, Fig. 2–1, is equal to VI, the voltage times the current. Then in another place you find that P is equal to , the square of the voltage di- vided by the resistance. In a third book you see that the power is equal to , the square of the current times the resistance! Which is it? Can they all be true? Even in this simple example you must know some algebra to make sense of such information. We will learn many new words in this chapter but since algebra is generalized arithmetic, some of what was said in Chapter 1 (such as rules of signs) will be re- peated here. I 2 R V 2 > R x n 3 2 x n (x)(x)(x)(x)...(x) ¯˚˚˚˘˚˚˚˙ x 2 (x)(x) x 2 3 2 (3)(3) 3 2 2 FIGURE 2–1 V I R
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