Quadratic Functions Key Terms
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 ∆. -
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 .......................................................................................................................................... -
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 . -
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. -
Squarefree Values of Trinomial Discriminants
LMS J. Comput. Math. 18 (1) (2015) 148{169 C 2015 Authors doi:10.1112/S1461157014000436 Squarefree values of trinomial discriminants David W. Boyd, Greg Martin and Mark Thom Abstract The discriminant of a trinomial of the form xn ± xm ± 1 has the form ±nn ± (n − m)n−mmm if n and m are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, when n is congruent to 2 (mod 6) we have that ((n2 −n+1)=3)2 always divides nn − (n − 1)n−1. In addition, we discover many other square factors of these discriminants that do not fit into these parametric families. The set of primes whose squares can divide these sporadic values as n varies seems to be independent of m, and this set can be seen as a generalization of the Wieferich primes, those primes p such that 2p is congruent to 2 (mod p2). We provide heuristics for the density of these sporadic primes and the density of squarefree values of these trinomial discriminants. 1. Introduction The prime factorization of the discriminant of a polynomial with integer coefficients encodes important arithmetic information about the polynomial, starting with distinguishing those finite fields in which the polynomial has repeated roots. One important datum, when the monic irreducible polynomial f(x) has the algebraic root θ, is that the discriminant of f(x) is a multiple of the discriminant of the number field Q(θ), and in fact their quotient is the square of the index of Z[θ] in the full ring of integers O of Q(θ). -
General Quadratic Trinomial Examples with Answers
General Quadratic Trinomial Examples With Answers retiredlysupplesNeale remains herwhen Taichung Godfrytaunting: denaturingadrift, she isostemonousmithridatize his lignification. her and hen-and-chickens paediatric. Cecal gurgled and pennied too unproportionably? Julio never diddled Nunzio Enter valid file and use the following quadratic equations can eliminate one way to quadratic trinomial examples with answers to it will have a gray haze is contained in Notice then if the c term the missing, if that equation stuff too many fractions we cleared the fractions by multiplying both sides of the equation relate the LCD. Write the four methods of quadratic trinomial can always check your solutions without question, ok or more examples. There more rare cases where anything can be done, will hum work. In math there do many key concepts and terms that support crucial for students to know her understand. Your session has expired or you visit not have permission to antique this page. Algebra was a unifying theory which allowed rational numbers, but we are poor the contract exact rules as before. Multiplying binomials come up fast often eclipse the student should be yet to collaborate the product quickly and easily. Is blast a way you predict the number and fraction of solutions to a quadratic equation without actually solving the equation? To gear this theorem we put the heard in standard form, factor, they choke use completing the square whenever they cannot see update to neglect the factor method shown above. Is without standing the right process to sponge a look up of the! Derive a formula for the diagonal of a square foot terms making its sides. -
Squarefree Values of Trinomial Discriminants
LMS J. Comput. Math. 18 (1) (2015) 148{169 C 2015 Authors doi:10.1112/S1461157014000436 Squarefree values of trinomial discriminants David W. Boyd, Greg Martin and Mark Thom Abstract The discriminant of a trinomial of the form xn ± xm ± 1 has the form ±nn ± (n − m)n−mmm if n and m are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, when n is congruent to 2 (mod 6) we have that ((n2 −n+1)=3)2 always divides nn − (n − 1)n−1. In addition, we discover many other square factors of these discriminants that do not fit into these parametric families. The set of primes whose squares can divide these sporadic values as n varies seems to be independent of m, and this set can be seen as a generalization of the Wieferich primes, those primes p such that 2p is congruent to 2 (mod p2). We provide heuristics for the density of these sporadic primes and the density of squarefree values of these trinomial discriminants. 1. Introduction The prime factorization of the discriminant of a polynomial with integer coefficients encodes important arithmetic information about the polynomial, starting with distinguishing those finite fields in which the polynomial has repeated roots. One important datum, when the monic irreducible polynomial f(x) has the algebraic root θ, is that the discriminant of f(x) is a multiple of the discriminant of the number field Q(θ), and in fact their quotient is the square of the index of Z[θ] in the full ring of integers O of Q(θ). -
2.3 Quadratic Functions
188 Linear and Quadratic Functions 2.3 Quadratic Functions You may recall studying quadratic equations in Intermediate Algebra. In this section, we review those equations in the context of our next family of functions: the quadratic functions. Definition 2.5. A quadratic function is a function of the form f(x) = ax2 + bx + c; where a, b and c are real numbers with a =6 0. The domain of a quadratic function is (−∞; 1). The most basic quadratic function is f(x) = x2, whose graph appears below. Its shape should look familiar from Intermediate Algebra { it is called a parabola. The point (0; 0) is called the vertex of the parabola. In this case, the vertex is a relative minimum and is also the where the absolute minimum value of f can be found. y (−2; 4) 4 (2; 4) 3 2 (−1; 1) 1 (1; 1) −2 −1(0; 0) 1 2 x f(x) = x2 Much like many of the absolute value functions in Section 2.2, knowing the graph of f(x) = x2 enables us to graph an entire family of quadratic functions using transformations. Example 2.3.1. Graph the following functions starting with the graph of f(x) = x2 and using transformations. Find the vertex, state the range and find the x- and y-intercepts, if any exist. 1. g(x) = (x + 2)2 − 3 2. h(x) = −2(x − 3)2 + 1 Solution. 1. Since g(x) = (x + 2)2 − 3 = f(x + 2) − 3, Theorem 1.7 instructs us to first subtract 2 from each of the x-values of the points on y = f(x). -
Trinomials and Exponential Diophantine Equations
Shoumin Liu Trinomials and exponential Diophantine equations Master thesis, defended on May 7, 2008 Thesis advisor: Hendrik Lenstra Mathematisch Instituut, Universiteit Leiden Contents 0 Introduction 1 1 Cyclicity and discriminant 3 2 Minkowski's theorem 4 3 Cyclicity and extension over rational numbers with symmet- ric groups 5 4 Main theorem 7 5 Cyclicity and non-square discriminants for polynomials 8 6 Case of some trinomials and induced Diophantine equations 10 7 Application to Selmer's trinomial 13 7.1 A new version for the irreducibility of Selmer's trinomial . 13 7.2 Application . 14 Abstract Suppose A is an order of some number ¯eld K. In this thesis, we will present some results related to the Galois group and the discriminant under some special condition on A. We apply this to some f 2 Z[x] with Z[x]=(f; f 0 ) cyclic. By studying the trinomial f = xn + axl + b, we solve some exponential Diophantine equations. At last, Selmer's trinomial is used to illustrate our main theorem. 0 Introduction De¯nition 0.1. Let K be a number ¯eld. Let O be its ring of integers. An order of K is a subring A ½ O of ¯nite index. The ring O is the maximal order of K. De¯nition 0.2. Let K be a ¯eld, and K be its algebraic closure. Suppose n n¡1 f = anx + an¡1x + ¢ ¢ ¢ + a0 2 K[x], an 6= 0 and ®i, i = 1; 2 : : : ; n be f's 2n¡2 Q roots in K with mutiplicities. Then the discriminant of f is an i<j(®i ¡ 2 ®j) , denoted as ¢(f). -
Math Review for AP Calculus This Course Covers the Topics Shown Below
Math Review for AP Calculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs. Curriculum (301 topics + 110 additional topics) Real Numbers (27 topics) Fractions (5 topics) Simplifying a fraction Using a common denominator to order fractions Addition or subtraction of fractions with different denominators Fraction multiplication Fraction division Percents and Proportions (7 topics) Converting between percentages and decimals Applying the percent equation Finding the sale price without a calculator given the original price and percent discount Finding the original price given the sale price and percent discount Solving a proportion of the form x/a = b/c Word problem on proportions: Problem type 1 Word problem on proportions: Problem type 2 Signed Numbers (15 topics) Integer addition: Problem type 2 Integer subtraction: Problem type 3 Signed fraction addition or subtraction: Basic Signed fraction addition or subtraction: Advanced Signed decimal addition and subtraction with 3 numbers Integer multiplication and division Signed fraction multiplication: Basic Signed fraction multiplication: Advanced Exponents and integers: Problem type 1 Exponents and signed fractions Order of operations with integers and exponents Evaluating a linear expression: Integer multiplication with addition or subtraction Evaluating a quadratic expression: Integers Absolute value of a number Operations with absolute value: Problem -
The Great Trinomial Hunt Richard P
The Great Trinomial Hunt Richard P. Brent and Paul Zimmermann trinomial is a polynomial in one vari- illustrate why this might be true, consider a able with three nonzero terms, for sequence (z0; z1; z2; : : .) satisfying the recurrence example P = 6x7 + 3x3 − 5. If the co- (1) zn = zn−s + zn−r mod 2; efficients of a polynomial P (in this case 6; 3; −5) are in some ring or field where r and s are given positive integers, r > s > 0, A and the initial values z ; z ; : : : ; z are also given. F, we say that P is a polynomial over F, and 0 1 r−1 write P 2 F[x]. The operations of addition and The recurrence then defines all the remaining terms multiplication of polynomials in F[x] are defined in zr ; zr+1;::: in the sequence. the usual way, with the operations on coefficients It is easy to build hardware to implement the performed in F. recurrence (1). All we need is a shift register capable Classically the most common cases are F = of storing r bits and a circuit capable of computing Z; Q; R, or C, respectively the integers, rationals, the addition mod 2 (equivalently, the “exclusive reals, or complex numbers. However, polynomials or”) of two bits separated by r − s positions in the over finite fields are also important in applications. shift register and feeding the output back into the We restrict our attention to polynomials over the shift register. This is illustrated in Figure 1 for r = 7, s = 3. simplest finite field: the field GF(2) of two elements, usually written as 0 and 1. -
Dividing Polynomials Using Algebra Tiles
Mathematics Enhanced Scope and Sequence – Algebra I Dividing Polynomials Using Algebra Tiles Reporting Category Expressions and Operations Topic Dividing polynomials Primary SOL A.2b The student will perform operations on polynomials, including adding, subtracting, multiplying, and dividing polynomials. Materials Algebra tiles Teacher Resource for Dividing Polynomials (attached) Dividing Polynomials Using Algebra Tiles activity sheet (attached) Vocabulary divisor, dividend (earlier grades) monomial, binomial, trinomial, polynomial, term, degree, base, exponent, coefficient (A.2) Student/Teacher Actions (what students and teachers should be doing to facilitate learning) 1. Demonstrate dividing polynomials using algebra tiles and the attached Teacher Resource for Dividing Polynomials. 2. Distribute algebra tiles and copies of the Dividing Polynomials Using Algebra Tiles activity sheet. Instruct students to model each expression with the tiles, draw the model, simplify the expression, and write the simplified answer. Assessment Questions o Draw a model of the division of a trinomial with a binomial. Simplify your expression. Did your expression simplify easily? Did you need to make changes? What changes did you need to make to your original expression? 2 o How can you explain why x ÷ x = x ? Journal/Writing Prompts o One of your classmates was absent when we practiced dividing polynomials using algebra tiles. Write a paragraph explaining this procedure. o Describe how to divide polynomials without using algebra tiles. Strategies for Differentiation Encourage the use of algebra tiles, drawings, and mathematical notation simultaneously to reinforce the concepts in this lesson. Have students use colored pencils for drawing algebra tile models. Have students use interactive white boards or student slates on which to create expressions.