5.2 Polynomial Functions 5.2 Polynomialfunctions
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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 Math´ematiques de Bordeaux, Universit´ede 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 Matem´aticas, 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 ∆. -
Lesson 1: Solutions to Polynomial Equations
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS Lesson 1: Solutions to Polynomial Equations Classwork Opening Exercise How many solutions are there to the equation 푥2 = 1? Explain how you know. Example 1: Prove that a Quadratic Equation Has Only Two Solutions over the Set of Complex Numbers Prove that 1 and −1 are the only solutions to the equation 푥2 = 1. Let 푥 = 푎 + 푏푖 be a complex number so that 푥2 = 1. a. Substitute 푎 + 푏푖 for 푥 in the equation 푥2 = 1. b. Rewrite both sides in standard form for a complex number. c. Equate the real parts on each side of the equation, and equate the imaginary parts on each side of the equation. d. Solve for 푎 and 푏, and find the solutions for 푥 = 푎 + 푏푖. Lesson 1: Solutions to Polynomial Equations S.1 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS Exercises Find the product. 1. (푧 − 2)(푧 + 2) 2. (푧 + 3푖)(푧 − 3푖) Write each of the following quadratic expressions as the product of two linear factors. 3. 푧2 − 4 4. 푧2 + 4 5. 푧2 − 4푖 6. 푧2 + 4푖 Lesson 1: Solutions to Polynomial Equations S.2 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. -
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 .......................................................................................................................................... -
Use of Iteration Technique to Find Zeros of Functions and Powers of Numbers
USE OF ITERATION TECHNIQUE TO FIND ZEROS OF FUNCTIONS AND POWERS OF NUMBERS Many functions y=f(x) have multiple zeros at discrete points along the x axis. The location of most of these zeros can be estimated approximately by looking at a graph of the function. Simple roots are the ones where f(x) passes through zero value with finite value for its slope while double roots occur for functions where its derivative also vanish simultaneously. The standard way to find these zeros of f(x) is to use the Newton-Raphson technique or a variation thereof. The procedure is to expand f(x) in a Taylor series about a starting point starting with the original estimate x=x0. This produces the iteration- 2 df (xn ) 1 d f (xn ) 2 0 = f (xn ) + (xn+1 − xn ) + (xn+1 − xn ) + ... dx 2! dx2 for which the function vanishes. For a function with a simple zero near x=x0, one has the iteration- f (xn ) xn+1 = xn − with x0 given f ′(xn ) For a double root one needs to keep three terms in the Taylor series to yield the iteration- − f ′(x ) + f ′(x)2 − 2 f (x ) f ′′(x ) n n n ∆xn+1 = xn+1 − xn = f ′′(xn ) To demonstrate how this procedure works, take the simple polynomial function- 2 4 f (x) =1+ 2x − 6x + x Its graph and the derivative look like this- There appear to be four real roots located near x=-2.6, -0.3, +0.7, and +2.2. None of these four real roots equal to an integer, however, they correspond to simple zeros near the values indicated. -
Factors, Zeros, and Solutions
Mathematics Instructional Plan – Algebra II Factors, Zeros, and Solutions Strand: Functions Topic: Exploring relationships among factors, zeros, and solutions Primary SOL: AII.8 The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Related SOL: AII.1d, AII.4b, AII.7b Materials Zeros and Factors activity sheet (attached) Matching Cards activity sheet (attached) Sorting Challenge activity sheet (attached) Graphing utility Colored paper Graph paper Vocabulary factor, fundamental theorem of algebra, imaginary numbers, multiplicity, polynomial, rate of change, real numbers, root, slope, solution, x-intercept, y-intercept, zeros of a function Student/Teacher Actions: What should students be doing? What should teachers be doing? Time: 90 minutes 1. Have students solve the equation 3x + 5 = 11. Then, have students set the left side of the equation equal to zero. Tell students to replace zero with y and consider the line y = 3x − 6. Instruct them to graph the line on graph paper. Review the terms slope, x- intercept, and y-intercept. Then, have them perform the same procedure (i.e., solve, set left side equal to zero, replace zero with y, and graph) for the equation −2x + 6 = 4. They should notice that in both cases, the x-intercept is the same as the solution to the equation. 2. Students explored factoring of zeros in Algebra I. Use the following practice problem to review: y = x2 + 4x + 3. Facilitate a discussion with students using the following questions: o “What is the first step in graphing the given line?” o “How do you find the factors of the line?” o “What role does zero play in graphing the line?” o “In how many ‘places’ did this line cross (intersect) the x-axis?” o “When the function crosses the x-axis, what is the value of y?” o “What is the significance of the x-intercepts of the graphs?” 3. -
Elementary Functions, Student's Text, Unit 21
DOCUMENT RESUME BD 135 629 SE 021 999 AUTHOR Allen, Frank B.; And Others TITLE Elementary Functions, Student's Text, Unit 21. INSTITUTION Stanford Univ., Calif. School Mathematics Study Group. SPONS AGENCY National Science Foundation, Washington, D.C. PUB DATE 61 NOTE 398p.; For related documents, see SE 021 987-022 002 and ED 130 870-877; Contains occasional light type EDRS PRICE MF-$0.83 HC-$20.75 Plus Postage. DESCRIPTO2S *Curriculum; Elementary Secondary Education; Instruction; *Instructional Materials; Mathematics Education; *Secondary School Mathematics; *Textbooks IDENTIFIERS *Functions (Mathematics); *School Mathematics Study Group ABSTRACT Unit 21 in the SMSG secondary school mathematics series is a student text covering the following topics in elementary functions: functions, polynomial functions, tangents to graphs of polynomial functions, exponential and logarithmic functions, and circular functions. Appendices discuss set notation, mathematical induction, significance of polynomials, area under a polynomial graph, slopes of area functions, the law of growth, approximation and computation of e raised to the x power, an approximation for ln x, measurement of triangles, trigonometric identities and equations, and calculation of sim x and cos x. (DT) *********************************************************************** Documents acquired by ERIC include many informal unpublished * materials not available from other sources. ERIC makes every effort * * to obtain the best copy available. Nevertheless, items of marginal * * reproducibility are often encountered and this affects the quality * * of the microfiche and hardcopy reproductions ERIC makes available * * via the ERIC Document Reproduction Service (EDRS). EDRS is not * responsible for the quality of the original document. Reproductions * * supplied by EDRS are the best that can be made from the original. -
Applications of Derivatives
5128_Ch04_pp186-260.qxd 1/13/06 12:35 PM Page 186 Chapter 4 Applications of Derivatives n automobile’s gas mileage is a function of many variables, including road surface, tire Atype, velocity, fuel octane rating, road angle, and the speed and direction of the wind. If we look only at velocity’s effect on gas mileage, the mileage of a certain car can be approximated by: m(v) ϭ 0.00015v 3 Ϫ 0.032v 2 ϩ 1.8v ϩ 1.7 (where v is velocity) At what speed should you drive this car to ob- tain the best gas mileage? The ideas in Section 4.1 will help you find the answer. 186 5128_Ch04_pp186-260.qxd 1/13/06 12:36 PM Page 187 Section 4.1 Extreme Values of Functions 187 Chapter 4 Overview In the past, when virtually all graphing was done by hand—often laboriously—derivatives were the key tool used to sketch the graph of a function. Now we can graph a function quickly, and usually correctly, using a grapher. However, confirmation of much of what we see and conclude true from a grapher view must still come from calculus. This chapter shows how to draw conclusions from derivatives about the extreme val- ues of a function and about the general shape of a function’s graph. We will also see how a tangent line captures the shape of a curve near the point of tangency, how to de- duce rates of change we cannot measure from rates of change we already know, and how to find a function when we know only its first derivative and its value at a single point. -
Optimization Algorithms on Matrix Manifolds
00˙AMS September 23, 2007 © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. Chapter Six Newton’s Method This chapter provides a detailed development of the archetypal second-order optimization method, Newton’s method, as an iteration on manifolds. We propose a formulation of Newton’s method for computing the zeros of a vector field on a manifold equipped with an affine connection and a retrac tion. In particular, when the manifold is Riemannian, this geometric Newton method can be used to compute critical points of a cost function by seeking the zeros of its gradient vector field. In the case where the underlying space is Euclidean, the proposed algorithm reduces to the classical Newton method. Although the algorithm formulation is provided in a general framework, the applications of interest in this book are those that have a matrix manifold structure (see Chapter 3). We provide several example applications of the geometric Newton method for principal subspace problems. 6.1 NEWTON’S METHOD ON MANIFOLDS In Chapter 5 we began a discussion of the Newton method and the issues involved in generalizing such an algorithm on an arbitrary manifold. Sec tion 5.1 identified the task as computing a zero of a vector field ξ on a Riemannian manifold equipped with a retraction R. The strategy pro M posed was to obtain a new iterate xk+1 from a current iterate xk by the following process. 1. Find a tangent vector ηk Txk such that the “directional derivative” of ξ along η is equal to ∈ ξ. -
Lesson 11 M1 ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M1 ALGEBRA II Lesson 11: The Special Role of Zero in Factoring Student Outcomes . Students find solutions to polynomial equations where the polynomial expression is not factored into linear factors. Students construct a polynomial function that has a specified set of zeros with stated multiplicity. Lesson Notes This lesson focuses on the first part of standard A-APR.B.3, identifying zeros of polynomials presented in factored form. Although the terms root and zero are interchangeable, for consistency only the term zero is used throughout this lesson and in later lessons. The second part of the standard, using the zeros to construct a rough graph of a polynomial function, is delayed until Lesson 14. The ideas that begin in this lesson continue in Lesson 19, in which students will be able to associate a zero of a polynomial function to a factor in the factored form of the associated polynomial as a consequence of the remainder theorem, and culminate in Lesson 39, in which students apply the fundamental theorem of algebra to factor polynomial expressions completely over the complex numbers. Classwork Scaffolding: Here is an alternative opening Opening Exercise (12 minutes) activity that may better illuminate the special role of Opening Exercise zero. ퟐ ퟐ Find all solutions to the equation (풙 + ퟓ풙 + ퟔ)(풙 − ퟑ풙 − ퟒ) = ퟎ. For each equation, list some possible values for 푥 and 푦. The main point of this opening exercise is for students to recognize and then formalize that the statement “If 푎푏 = 0, then 푎 = 0 or 푏 = 0” applies not only when 푎 and 푏 are 푥푦 = 10, 푥푦 = 1, numbers or linear functions (which we used when solving a quadratic equation), but also 푥푦 = −1, 푥푦 = 0 applies to cases where 푎 and 푏 are polynomial functions of any degree. -
Real Zeros of Polynomial Functions
333353_0200.qxp 1/11/07 2:00 PM Page 91 Polynomial and Chapter 2 Rational Functions y y y 2.1 Quadratic Functions 2 2 2 2.2 Polynomial Functions of Higher Degree x x x −44−2 −44−2 −44−2 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra Polynomial and rational functions are two of the most common types of functions 2.6 Rational Functions and used in algebra and calculus. In Chapter 2, you will learn how to graph these types Asymptotes of functions and how to find the zeros of these functions. 2.7 Graphs of Rational Functions 2.8 Quadratic Models David Madison/Getty Images Selected Applications Polynomial and rational functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. ■ Automobile Aerodynamics, Exercise 58, page 101 ■ Revenue, Exercise 93, page 114 ■ U.S. Population, Exercise 91, page 129 ■ Impedance, Exercises 79 and 80, page 138 ■ Profit, Exercise 64, page 145 ■ Data Analysis, Exercises 41 and 42, page 154 ■ Wildlife, Exercise 43, page 155 ■ Comparing Models, Exercise 85, page 164 ■ Media, Aerodynamics is crucial in creating racecars.Two types of racecars designed and built Exercise 18, page 170 by NASCAR teams are short track cars, as shown in the photo, and super-speedway (long track) cars. Both types of racecars are designed either to allow for as much downforce as possible or to reduce the amount of drag on the racecar. 91 333353_0201.qxp 1/11/07 2:02 PM Page 92 92 Chapter 2 Polynomial and Rational Functions 2.1 Quadratic Functions The Graph of a Quadratic Function What you should learn Analyze graphs of quadratic functions. -
Root-Finding 3.1 Physical Problems
Physics 200 Lecture 3 Root-Finding Lecture 3 Physics 200 Laboratory Monday, February 14th, 2011 The fundamental question answered by this week's lab work will be: Given a function F (x), find some/all of the values xi for which F (xi) = 0. It's a f g modest goal, and we will use a simple method to solve the problem. But, as we shall see, there are a wide range of physical problems that have, at their heart, just such a question. We'll start in the simplest, polynomial setting, and work our way up to the \shooting" method. 3.1 Physical Problems We'll set up some direct applications of root-finding with familiar physical examples, and then shift gears and define a numerical root-finding routine that can be used to solve a very different set of problems. 3.1.1 Orbital Motion In two-dimensions, with a spherically symmetric potential (meaning here that V (x; y; z) = V (r), a function of a single variable, r x2 + y2 + z2, ≡ the distance to the origin) we can use circular coordinates to write the total p energy of a test particle moving under the influence of this potential as 1 E = m r_2 + r2 φ_2 + V (r): (3.1) 2 Conservation of momentum tells us that the z-component of angular mo- 2 mentum is conserved, with Lz = (r p) = m r φ_, so we can rewrite the × z 1 of 13 3.1. PHYSICAL PROBLEMS Lecture 3 energy as: 1 1 L2 E = m r_2 + z + V (r) (3.2) 2 2 m r2 ≡U(r) where U(r) defines an “effective potential"| { we{z have} turned a two-dimensional problem into a one-dimensional problem for the coordinate r, and an effec- tive potential that governs the motion in this setting. -
Polynomials Remember from 7-1: a Monomial Is a Number, a Variable, Or a Product of Numbers and Variables with Whole-Number Exponents
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.