Polynomials Remember from 7-1: a Monomial Is a Number, a Variable, Or a Product of Numbers and Variables with Whole-Number Exponents

Total Page：16

File Type：pdf, Size：1020Kb

Notes 7-3: Polynomials Remember from 7-1: A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents. A monomial may be a constant or a single variable. I. Identifying Polynomials A polynomial is a monomial or a sum or difference of monomials. Some polynomials have special names. A binomial is the sum of two monomials. A trinomial is the sum of three monomials. • Example: State whether the expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. Expression Polynomial? Monomial, Binomial, or Trinomial? 2x - 3yz Yes, 2x - 3yz = 2x + (-3yz), the binomial sum of two monomials 8n3+5n-2 No, 5n-2 has a negative None of these exponent, so it is not a monomial -8 Yes, -8 is a real number Monomial 4a2 + 5a + a + 9 Yes, the expression simplifies Monomial to 4a2 + 6a + 9, so it is the sum of three monomials II. Degrees and Leading Coefficients The terms of a polynomial are the monomials that are being added or subtracted. The degree of a polynomial is the degree of the term with the greatest degree. The leading coefficient is the coefficient of the variable with the highest degree. Find the degree and leading coefficient of each polynomial Polynomial Terms Degree Leading Coefficient 5n2 5n 2 2 5 -4x3 + 3x2 + 5 -4x2, 3x2, 3 -4 5 -a4-1 -a4, -1 4 -1 III. Ordering the terms of a polynomial The terms of a polynomial may be written in any order. However, the terms of a polynomial are usually arranged so that the powers of one variable are in descending (decreasing, large to small) order. Examples: Arrange the terms of each polynomial so that the powers of x are in descending order. A. 7x2 + 2x4 - 11 B. 2xy3 + y2 + 5x3 – 3x2y 2x4 + 7x2 – 11x 2x1y3 + y2 + 5x3 – 3x2y1 5x3 -3x2y + 2xy3 +y2 Arrange the terms of each polynomial so that the powers of x are in descending order. C. 6x2 + 5 – 8x -2x3 -2x3 + 6x2– 8x + 5 D. 3a3x2 – a4 + 4ax5 + 9a2x 4ax5 + 3a3x2 + 9a2x – a4 You Try! Arrange the terms of each polynomial so that the powers of x are in descending order. 2 4 4 2 3 5 2 E. 3x y + 2x y – 4x y + x – y x5+ 2x4y2– 4x3y + 3x2y4 -y2 .

Copy Link

Recommended publications

Discriminant Analysis
Chapter 5 Discriminant Analysis This chapter considers discriminant analysis: given p measurements w, we want to correctly classify w into one of G groups or populations. The max- imum likelihood, Bayesian, and Fisher’s discriminant rules are used to show why methods like linear and quadratic discriminant analysis can work well for a wide variety of group distributions. 5.1 Introduction Definition 5.1. In supervised classification, there are G known groups and m test cases to be classified. Each test case is assigned to exactly one group based on its measurements wi. Suppose there are G populations or groups or classes where G 2. Assume ≥ that for each population there is a probability density function (pdf) fj(z) where z is a p 1 vector and j =1,...,G. Hence if the random vector x comes × from population j, then x has pdf fj (z). Assume that there is a random sam- ple of nj cases x1,j,..., xnj ,j for each group. Let (xj , Sj ) denote the sample mean and covariance matrix for each group. Let wi be a new p 1 (observed) random vector from one of the G groups, but the group is unknown.× Usually there are many wi, and discriminant analysis (DA) or classification attempts to allocate the wi to the correct groups. The w1,..., wm are known as the test data. Let πk = the (prior) probability that a randomly selected case wi belongs to the kth group. If x1,1..., xnG,G are a random sample of cases from G the collection of G populations, thenπ ˆk = nk/n where n = i=1 ni.
[Show full text]
Solving Quadratic Equations by Factoring
5.2 Solving Quadratic Equations by Factoring Goals p Factor quadratic expressions and solve quadratic equations by factoring. p Find zeros of quadratic functions. Your Notes VOCABULARY Binomial An expression with two terms, such as x ϩ 1 Trinomial An expression with three terms, such as x2 ϩ x ϩ 1 Factoring A process used to write a polynomial as a product of other polynomials having equal or lesser degree Monomial An expression with one term, such as 3x Quadratic equation in one variable An equation that can be written in the form ax2 ϩ bx ϩ c ϭ 0 where a 0 Zero of a function A number k is a zero of a function f if f(k) ϭ 0. Example 1 Factoring a Trinomial of the Form x2 ؉ bx ؉ c Factor x2 Ϫ 9x ϩ 14. Solution You want x2 Ϫ 9x ϩ 14 ϭ (x ϩ m)(x ϩ n) where mn ϭ 14 and m ϩ n ϭ Ϫ9 . Factors of 14 1, Ϫ1, Ϫ 2, Ϫ2, Ϫ (m, n) 14 14 7 7 Sum of factors Ϫ Ϫ m ؉ n) 15 15 9 9) The table shows that m ϭ Ϫ2 and n ϭ Ϫ7 . So, x2 Ϫ 9x ϩ 14 ϭ ( x Ϫ 2 )( x Ϫ 7 ). Lesson 5.2 • Algebra 2 Notetaking Guide 97 Your Notes Example 2 Factoring a Trinomial of the Form ax2 ؉ bx ؉ c Factor 2x2 ϩ 13x ϩ 6. Solution You want 2x2 ϩ 13x ϩ 6 ϭ (kx ϩ m)(lx ϩ n) where k and l are factors of 2 and m and n are ( positive ) factors of 6 .
[Show full text]
Alicia Dickenstein
A WORLD OF BINOMIALS ——————————– ALICIA DICKENSTEIN Universidad de Buenos Aires FOCM 2008, Hong Kong - June 21, 2008 A. Dickenstein - FoCM 2008, Hong Kong – p.1/39 ORIGINAL PLAN OF THE TALK BASICS ON BINOMIALS A. Dickenstein - FoCM 2008, Hong Kong – p.2/39 ORIGINAL PLAN OF THE TALK BASICS ON BINOMIALS COUNTING SOLUTIONS TO BINOMIAL SYSTEMS A. Dickenstein - FoCM 2008, Hong Kong – p.2/39 ORIGINAL PLAN OF THE TALK BASICS ON BINOMIALS COUNTING SOLUTIONS TO BINOMIAL SYSTEMS DISCRIMINANTS (DUALS OF BINOMIAL VARIETIES) A. Dickenstein - FoCM 2008, Hong Kong – p.2/39 ORIGINAL PLAN OF THE TALK BASICS ON BINOMIALS COUNTING SOLUTIONS TO BINOMIAL SYSTEMS DISCRIMINANTS (DUALS OF BINOMIAL VARIETIES) BINOMIALS AND RECURRENCE OF HYPERGEOMETRIC COEFFICIENTS A. Dickenstein - FoCM 2008, Hong Kong – p.2/39 ORIGINAL PLAN OF THE TALK BASICS ON BINOMIALS COUNTING SOLUTIONS TO BINOMIAL SYSTEMS DISCRIMINANTS (DUALS OF BINOMIAL VARIETIES) BINOMIALS AND RECURRENCE OF HYPERGEOMETRIC COEFFICIENTS BINOMIALS AND MASS ACTION KINETICS DYNAMICS A. Dickenstein - FoCM 2008, Hong Kong – p.2/39 RESCHEDULED PLAN OF THE TALK BASICS ON BINOMIALS - How binomials “sit” in the polynomial world and the main “secrets” about binomials A. Dickenstein - FoCM 2008, Hong Kong – p.3/39 RESCHEDULED PLAN OF THE TALK BASICS ON BINOMIALS - How binomials “sit” in the polynomial world and the main “secrets” about binomials COUNTING SOLUTIONS TO BINOMIAL SYSTEMS - with a touch of complexity A. Dickenstein - FoCM 2008, Hong Kong – p.3/39 RESCHEDULED PLAN OF THE TALK BASICS ON BINOMIALS - How binomials “sit” in the polynomial world and the main “secrets” about binomials COUNTING SOLUTIONS TO BINOMIAL SYSTEMS - with a touch of complexity (Mixed) DISCRIMINANTS (DUALS OF BINOMIAL VARIETIES) and an application to real roots A.
[Show full text]
The Diamond Method of Factoring a Quadratic Equation
The Diamond Method of Factoring a Quadratic Equation Important: Remember that the first step in any factoring is to look at each term and factor out the greatest common factor. For example: 3x2 + 6x + 12 = 3(x2 + 2x + 4) AND 5x2 + 10x = 5x(x + 2) If the leading coefficient is negative, always factor out the negative. For example: -2x2 - x + 1 = -1(2x2 + x - 1) = -(2x2 + x - 1) Using the Diamond Method: Example 1 2 Factor 2x + 11x + 15 using the Diamond Method. +30 Step 1: Multiply the coefficient of the x2 term (+2) and the constant (+15) and place this product (+30) in the top quarter of a large “X.” Step 2: Place the coefficient of the middle term in the bottom quarter of the +11 “X.” (+11) Step 3: List all factors of the number in the top quarter of the “X.” +30 (+1)(+30) (-1)(-30) (+2)(+15) (-2)(-15) (+3)(+10) (-3)(-10) (+5)(+6) (-5)(-6) +30 Step 4: Identify the two factors whose sum gives the number in the bottom quarter of the “x.” (5 ∙ 6 = 30 and 5 + 6 = 11) and place these factors in +5 +6 the left and right quarters of the “X” (order is not important). +11 Step 5: Break the middle term of the original trinomial into the sum of two terms formed using the right and left quarters of the “X.” That is, write the first term of the original equation, 2x2 , then write 11x as + 5x + 6x (the num bers from the “X”), and finally write the last term of the original equation, +15 , to get the following 4-term polynomial: 2x2 + 11x + 15 = 2x2 + 5x + 6x + 15 Step 6: Factor by Grouping: Group the first two terms together and the last two terms together.
[Show full text]
A Review of Some Basic Mathematical Concepts and Differential Calculus
A Review of Some Basic Mathematical Concepts and Diﬀerential Calculus Kevin Quinn Assistant Professor Department of Political Science and The Center for Statistics and the Social Sciences Box 354320, Padelford Hall University of Washington Seattle, WA 98195-4320 October 11, 2002 1 Introduction These notes are written to give students in CSSS/SOC/STAT 536 a quick review of some of the basic mathematical concepts they will come across during this course. These notes are not meant to be comprehensive but rather to be a succinct treatment of some of the key ideas. The notes draw heavily from Apostol (1967) and Simon and Blume (1994). Students looking for a more detailed presentation are advised to see either of these sources. 2 Preliminaries 2.1 Notation 1 1 R or equivalently R denotes the real number line. R is sometimes referred to as 1-dimensional 2 Euclidean space. 2-dimensional Euclidean space is represented by R , 3-dimensional space is rep- 3 k resented by R , and more generally, k-dimensional Euclidean space is represented by R . 1 1 Suppose a ∈ R and b ∈ R with a < b. 1 A closed interval [a, b] is the subset of R whose elements are greater than or equal to a and less than or equal to b. 1 An open interval (a, b) is the subset of R whose elements are greater than a and less than b. 1 A half open interval [a, b) is the subset of R whose elements are greater than or equal to a and less than b.
[Show full text]
Factoring Polynomials
2/13/2013 Chapter 13 § 13.1 Factoring The Greatest Common Polynomials Factor Chapter Sections Factors 13.1 – The Greatest Common Factor Factors (either numbers or polynomials) 13.2 – Factoring Trinomials of the Form x2 + bx + c When an integer is written as a product of 13.3 – Factoring Trinomials of the Form ax 2 + bx + c integers, each of the integers in the product is a factor of the original number. 13.4 – Factoring Trinomials of the Form x2 + bx + c When a polynomial is written as a product of by Grouping polynomials, each of the polynomials in the 13.5 – Factoring Perfect Square Trinomials and product is a factor of the original polynomial. Difference of Two Squares Factoring – writing a polynomial as a product of 13.6 – Solving Quadratic Equations by Factoring polynomials. 13.7 – Quadratic Equations and Problem Solving Martin-Gay, Developmental Mathematics 2 Martin-Gay, Developmental Mathematics 4 1 2/13/2013 Greatest Common Factor Greatest Common Factor Greatest common factor – largest quantity that is a Example factor of all the integers or polynomials involved. Find the GCF of each list of numbers. 1) 6, 8 and 46 6 = 2 · 3 Finding the GCF of a List of Integers or Terms 8 = 2 · 2 · 2 1) Prime factor the numbers. 46 = 2 · 23 2) Identify common prime factors. So the GCF is 2. 3) Take the product of all common prime factors. 2) 144, 256 and 300 144 = 2 · 2 · 2 · 3 · 3 • If there are no common prime factors, GCF is 1. 256 = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 300 = 2 · 2 · 3 · 5 · 5 So the GCF is 2 · 2 = 4.
[Show full text]
Trinomials, Singular Moduli and Riffaut's Conjecture
Trinomials, singular moduli and Riffaut’s conjecture Yuri Bilu,a Florian Luca,b Amalia Pizarro-Madariagac August 30, 2021 Abstract Riffaut [22] conjectured that a singular modulus of degree h ≥ 3 cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH and obtain partial unconditional results. Contents 1 Introduction 1 2 Generalities on singular moduli 4 3 Suitable integers 5 4 Rootsoftrinomialsandtheprincipalinequality 8 5 Suitable integers for trinomial discriminants 11 6 Small discriminants 13 7 Structure of trinomial discriminants 20 8 Primality of suitable integers 24 9 A conditional result 25 10Boundingallbutonetrinomialdiscriminants 29 11 The quantities h(∆), ρ(∆) and N(∆) 35 12 The signature theorem 40 1 Introduction arXiv:2003.06547v2 [math.NT] 26 Aug 2021 A singular modulus is the j-invariant of an elliptic curve with complex multi- plication. Given a singular modulus x we denote by ∆x the discriminant of the associated imaginary quadratic order. We denote by h(∆) the class number of aInstitut de Mathématiques de Bordeaux, Universitéde Bordeaux & CNRS; partially sup- ported by the MEC CONICYT Project PAI80160038 (Chile), and by the SPARC Project P445 (India) bSchool of Mathematics, University of the Witwatersrand; Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah; Centro de Ciencias Matem- aticas, UNAM, Morelia; partially supported by the CNRS “Postes rouges” program cInstituto de Matemáticas, Universidad de Valpara´ıso; partially supported by the Ecos / CONICYT project ECOS170022 and by the MATH-AmSud project NT-ACRT 20-MATH-06 1 the imaginary quadratic order of discriminant ∆. Recall that two singular mod- uli x and y are conjugate over Q if and only if ∆x = ∆y, and that there are h(∆) singular moduli of a given discriminant ∆.
[Show full text]
Factoring Polynomials
EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: — Set the factors of a polynomial equation (as opposed to an expression) equal to zero in order to solve for a variable: Example: To solve ,; , The flowchart below illustrates a sequence of steps for factoring polynomials. First, always factor out the Greatest Common Factor (GCF), if one exists. Is the equation a Binomial or a Trinomial? 1 Prime polynomials cannot be factored Yes No using integers alone. The Sum of Squares and the Four or more quadratic factors Special Cases? terms of the Sum and Difference of Binomial Trinomial Squares are (two terms) (three terms) Factor by Grouping: always Prime. 1. Group the terms with common factors and factor 1. Difference of Squares: out the GCF from each Perfe ct Square grouping. 1 , 3 Trinomial: 2. Sum of Squares: 1. 2. Continue factoring—by looking for Special Cases, 1 , 2 2. 3. Difference of Cubes: Grouping, etc.—until the 3 equation is in simplest form FYI: A Sum of Squares can 1 , 2 (or all factors are Prime). 4. Sum of Cubes: be factored using imaginary numbers if you rewrite it as a Difference of Squares: — 2 Use S.O.A.P to No Special √1 √1 Cases remember the signs for the factors of the 4 Completing the Square and the Quadratic Formula Sum and Difference Choose: of Cubes: are primarily methods for solving equations rather 1. Factor by Grouping than simply factoring expressions.
[Show full text]
Chapter 8: Exponents and Polynomials
Chapter 8 CHAPTER 8: EXPONENTS AND POLYNOMIALS Chapter Objectives By the end of this chapter, students should be able to: Simplify exponential expressions with positive and/or negative exponents Multiply or divide expressions in scientific notation Evaluate polynomials for specific values Apply arithmetic operations to polynomials Apply special-product formulas to multiply polynomials Divide a polynomial by a monomial or by applying long division CHAPTER 8: EXPONENTS AND POLYNOMIALS ........................................................................................ 211 SECTION 8.1: EXPONENTS RULES AND PROPERTIES ........................................................................... 212 A. PRODUCT RULE OF EXPONENTS .............................................................................................. 212 B. QUOTIENT RULE OF EXPONENTS ............................................................................................. 212 C. POWER RULE OF EXPONENTS .................................................................................................. 213 D. ZERO AS AN EXPONENT............................................................................................................ 214 E. NEGATIVE EXPONENTS ............................................................................................................. 214 F. PROPERTIES OF EXPONENTS .................................................................................................... 215 EXERCISE ..........................................................................................................................................
[Show full text]
A Saturation Algorithm for Homogeneous Binomial Ideals
A Saturation Algorithm for Homogeneous Binomial Ideals Deepanjan Kesh and Shashank K Mehta Indian Institute of Technology, Kanpur - 208016, India, fdeepkesh,[email protected] Abstract. Let k[x1; : : : ; xn] be a polynomial ring in n variables, and let I ⊂ k[x1; : : : ; xn] be a homogeneous binomial ideal. We describe a fast 1 algorithm to compute the saturation, I :(x1 ··· xn) . In the special case when I is a toric ideal, we present some preliminary results comparing our algorithm with Project and Lift by Hemmecke and Malkin. 1 Introduction 1.1 Problem Description Let k[x1; : : : ; xn] be a polynomial ring in n variables over the field k, and let I ⊂ k[x1; : : : ; xn] be an ideal. Ideals are said to be homogeneous, if they have a basis consisting of homogeneous polynomials. Binomials in this ring are defined as polynomials with at most two terms [5]. Thus, a binomial is a polynomial of the form cxα+dxβ, where c; d are arbitrary coefficients. Pure difference binomials are special cases of binomials of the form xα − xβ. Ideals with a binomial basis are called binomial ideals. Toric ideals, the kernel of a specific kind of polynomial ring homomorphisms, are examples of pure difference binomial ideals. Saturation of an ideal, I, by a polynomial f, denoted by I : f, is defined as the ideal I : f = hf g 2 k[x1; : : : ; xn]: f · g 2 I gi: Similarly, I : f 1 is defined as 1 a I : f = hf g 2 k[x1; : : : ; xn]: 9a 2 N; f · g 2 I gi: 1 We describe a fast algorithm to compute the saturation, I :(x1 ··· xn) , of a homogeneous binomial ideal I.
[Show full text]
Linear Gaps Between Degrees for the Polynomial Calculus Modulo Distinct Primes
Linear Gaps Between Degrees for the Polynomial Calculus Modulo Distinct Primes Sam Buss1;2 Dima Grigoriev Department of Mathematics Computer Science and Engineering Univ. of Calif., San Diego Pennsylvania State University La Jolla, CA 92093-0112 University Park, PA 16802-6106 [email protected] [email protected] Russell Impagliazzo1;3 Toniann Pitassi1;4 Computer Science and Engineering Computer Science Univ. of Calif., San Diego University of Arizona La Jolla, CA 92093-0114 Tucson, AZ 85721-0077 [email protected] [email protected] Abstract e±cient search algorithms and in part by the desire to extend lower bounds on proposition proof complexity This paper gives nearly optimal lower bounds on the to stronger proof systems. minimum degree of polynomial calculus refutations of The Nullstellensatz proof system is a propositional Tseitin's graph tautologies and the mod p counting proof system based on Hilbert's Nullstellensatz and principles, p 2. The lower bounds apply to the was introduced in [1]. The polynomial calculus (PC) ¸ polynomial calculus over ¯elds or rings. These are is a stronger propositional proof system introduced the ¯rst linear lower bounds for polynomial calculus; ¯rst by [4]. (See [8] and [3] for subsequent, more moreover, they distinguish linearly between proofs over general treatments of algebraic proof systems.) In the ¯elds of characteristic q and r, q = r, and more polynomial calculus, one begins with an initial set of 6 generally distinguish linearly the rings Zq and Zr where polynomials and the goal is to prove that they cannot q and r do not have the identical prime factors.
[Show full text]
5.2 Polynomial Functions 5.2 Polynomialfunctions
5.2 Polynomial Functions 5.2 PolynomialFunctions This is part of 5.1 in the current text In this section, you will learn about the properties, characteristics, and applications of polynomial functions. Upon completion you will be able to: • Recognize the degree, leading coefficient, and end behavior of a given polynomial function. • Memorize the graphs of parent polynomial functions (linear, quadratic, and cubic). • State the domain of a polynomial function, using interval notation. • Define what it means to be a root/zero of a function. • Identify the coefficients of a given quadratic function and he direction the corresponding graph opens. • Determine the verte$ and properties of the graph of a given quadratic function (domain, range, and minimum/maximum value). • Compute the roots/zeroes of a quadratic function by applying the quadratic formula and/or by factoring. • !&etch the graph of a given quadratic function using its properties. • Use properties of quadratic functions to solve business and social science problems. DescribingPolynomialFunctions ' polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is the product of a number, called the coefficient of the term, and a variable raised to a non-negative integer (a natural number) power. Definition ' polynomial function is a function of the form n n ) * f(x =a n x +a n ) x − +...+a * x +a ) x+a +, − wherea ,a ,...,a are real numbers andn ) is a natural number. � + ) n ≥ In a previous chapter we discussed linear functions, f(x =mx=b , and the equation of a horizontal line, y=b .
[Show full text]