Copyrighted Material. Chapter 1 DEGREE OF A CURVE


 Austen Cain
 6 years ago
 Views:
Transcription
1 Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two different definitions of the degree of an algebraic curve. Our job in the next few chapters will be to show that these two different definitions, suitably interpreted, agree. During our journey of discovery, we will often use elliptic curves as typical examples of algebraic curves. Often, we ll use y 2 = x 3 x or y 2 = x 3 + 3x as our examples. 1. Greek Mathematics In this chapter, we will begin exploring the concept of the degree of an algebraic curve that is, a curve that can be defined by polynomial equations. We will see that a circle has degree 2. The ancient Greeks also studied lines and planes, which have degree 1. Euclid limited himself to a straightedge and compass, which can create curves only of degrees 1 and 2. A primer of these results may be found in the Elements (Euclid, 1956). Because 1 and 2 are the lowest degrees, the Greeks were very successful in this part of algebraic geometry. (Of course, they thought only of geometry, not of algebra.) Greek mathematicians also invented methods that constructed higher degree curves, and even nonalgebraic curves, such as spirals. (The latter cannot be defined using polynomial equations.) They were aware that
2 14 CHAPTER 1 Figure 1.1. Three curves these tools enabled them to go beyond what they could do with straightedge and compass. In particular, they solved the problems of doubling the cube and trisecting angles. Both of these are problems of degree 3, the same degree as the elliptic curves that are the main subject of this book. Doubling the cube requires solving the equation x 3 = 2, which is clearly degree 3. Trisecting an angle involves finding the intersection of a circle and a hyperbola, which also turns out to be equivalent to solving an equation of degree 3. See Thomas (1980, pp , pp , and the footnotes) and Heath (1981, pp ) for details of these constructions. Squaring the circle is beyond any tool that can construct only algebraic curves; the ultimate reason is that π is not the root of any polynomial with integer coefficients. As in the previous two paragraphs, we will see that the degree is a useful way of arranging algebraic and geometric objects in a hierarchy. Often, the degree coincides with the level of difficulty in understanding them. 2. Degree We have a feeling that some shapes are simpler than others. For example, a line is simpler than a circle, and a circle is simpler than a cubic curve; see figure 1.1 You might argue as to whether a cubic curve is simpler than a sine wave or not. Once algebra has been developed, we can follow the lead of French mathematician René Descartes ( ), and try writing down algebraic equations whose solution sets yield the curves in which we are interested. For example, the line, circle, and cubic curve in figure 1.1 have equations x + y = 0, x 2 + y 2 = 1, and y 2 = x 3 x 1, respectively. On the other hand, as we will see, the sine curve cannot be described by an algebraic equation.
3 DEGREE OF A CURVE 15 Figure 1.2. y 2 = x 3 x Our typical curve with degree 3 has the equation y 2 = x 3 x.aswecan see in figure 1.2, the graph of this equation has two pieces. We can extend the concept of equations to higher dimensions also. For example a sphere of radius r can be described by the equation x + y + z = r. (1.1) A certain line in 3dimensional space is described by the pair of simultaneous equations x + y + z = 5 x z = 0. (1.2) The solution set to a system of simultaneous equations is the set of all ways that we can assign numbers to the variables and make all the equations in the system true at the same time. For example, in the equation of the sphere (which is a system of simultaneous equations containing only one equation), the solution set is the set of all triples of the form J (x, y, z) = (a, b, ± r 2 a 2 b 2 ). This means: To get a single element of the solution set, you pick any two numbers a and b, and youset x = a, y = b, and z = r 2 a 2 b 2 or z = r 2 a 2 b 2. (If you don t want to use complex numbers, and you
4 16 CHAPTER 1 only want to look at the real sphere, then you should make sure that a 2 + b 2 r 2.) Similarly, the solution set to the pair of linear equations in (1.2) can be described as the set of all triples (x, y, z) = (t,5 2t, t), where t can be any number. As for our prototypical cubic curve y 2 = x 3 x, we see that its solution set includes (0, 0), (1, 0), and ( 1, 0), but it is difficult to see what the entire set of solutions is. In this book, we will consider mostly systems of algebraic equations. That means by definition that both sides of the equation have to be polynomial expressions in the variables. The solution sets to such systems are called algebraic varieties. The study of algebraic geometry, which was initiated by Descartes, is the study of these solution sets. Since we can restrict our attention to solutions that are integers, or rational numbers, if we want to, a large chunk of number theory also falls under the rubric of algebraic geometry. Some definitions: A polynomial in one or several variables is an algebraic expression that involves only addition, subtraction, and multiplication of the constants and variables. (Division by a variable is not permitted.) Therefore, each variable might be raised to a positive integral power, but not a negative or a fractional power. A monomial is a polynomial involving only multiplication, but no addition or subtraction. The degree of a monomial is the sum of the powers of the variables that occur in the monomial. The degree of the zero monomial is undefined. For example, the monomial 3xy 2 z 5 has degree 8, because = 8. The degree of a polynomial is the largest degree of any of the monomials in the polynomial. We assume that the polynomial is written without any terms that can be combined or cancelled. For example, the polynomial 3xy 2 z 5 + 2x 3 z 3 + xyz + 5 has degree 8, because the other terms have degrees 6, 3, and 0, which are all
5 DEGREE OF A CURVE 17 smaller than 8. The polynomial y 5 x 3 y xy has degree 3, because we first must cancel the two y 5 terms before computing the degree. The polynomial 2x 7 y 2 x 7 y 2 + xy 1 x 7 y 2 has degree 2, because it is really the polynomial xy 1. If we have an algebraic variety defined by a system of equations of the form some polynomial = some other polynomial, we say that the variety has degree d if the largest degree of any polynomial appearing in the system of equations is d.again,we assume that the system of equations cannot be simplified into an equivalent system of equations with smaller degree. EXERCISE: What is the degree of the equation for a sphere in (1.1)? What is the degree of the system of equations for the line in (1.2)? SOLUTION: The degree of a sphere is 2. The degree of a line is 1. Now suppose we have a geometric curve or shape. How can we tell what its degree is if we are not given its equation(s)? Or maybe it isn t given by any system of algebraic equations? We re not going to give a general answer to this question in this book, but we will explain the basic idea in the case of single equations. Let s start by recalling another definition. DEFINITION: Suppose that p(x) is a polynomial. If b is a number so that p(b) = 0, then b is a root of the polynomial p(x). The basic idea is that we will use lines as probes to tell us the degree of a polynomial. This method is based on a very important fact about polynomials: Suppose you have a polynomial f (x) = a n x n + a n 1 x n 1 + +a 0 where a n = 0, so that f (x) has degree n, where n 1. Suppose that b is a root of f (x). Then if you divide x b into f (x), it will go in evenly, without remainder.
6 18 CHAPTER 1 For example, if f (x) = x 3 x 6, you can check that f (2) = 0. Now divide x 2 into x 3 x 6 using long division: x 2 + 2x + 3 x 2) x 3 x 6 x 3 2x 2 2x 2 x 2x 2 4x 3x 6 3x 6 0 We get a quotient of x 2 + 2x + 3, without remainder. Another way to say this is that x 3 x 6 = (x 2 + 2x + 3)(x 2). We can prove our assertion in general: Suppose you divide x b into f (x) and get the quotient q(x) with remainder r. The remainder r in polynomial division is always a polynomial of degree less than the divisor. The divisor x b has degree 1, so the remainder must have degree 0. In other words, r is some number. We then take the true statement f (x) = q(x)(x b) + r, and set x = b to get f (b) = q(b)(b b) + r. Since f (b) = 0and b b = 0, we deduce that r = 0. So there was no remainder, as we claimed. Now this little fact has the momentous implication that the number of roots of a polynomial f (x) cannot be greater than its degree. Why not? Suppose f (x) had the roots b 1,..., b k, all different from each other. Then you can keep factoring out the various x b i s and get that f (x) = q(x)(x b 1 )(x b 2 ) (x b k ) for some nonzero polynomial q(x). Now multiply out all those factors on the righthand side. The highest power of x you get must be at least k.since the highest power of x on the lefthand side is the degree of f (x), we know that k is no greater than the degree of f (x). Next, we interpret geometrically what it means for b to be a root of the polynomial f (x)ofdegree n. Look at the graph G of y = f (x). It is a picture in the Cartesian plane of all pairs (x, y) where y = f (x). Now look at the
7 DEGREE OF A CURVE 19 y G x Figure 1.3. y = x(x 1)(x + 1) y x Figure 1.4. y = sin x graph of y = 0. It is a horizontal straight line L consisting of all pairs (x, y) where y = 0and x can be anything. OK, now look at the intersection of the graph of f (x) and the line L. Which points are in the intersection? They are exactly the points (b,0) where 0 = f (b). That means that the xcoordinates of the points of intersection, the b s, are exactly the roots of f (x). There can be at most n of these roots. This means that the line L can hit the curve G in at most n points. Figure 1.3 contains an example using the function x 3 x = x(x 1)(x + 1), which is the righthand side of our continuing example cubic curve y 2 = x 3 x. On the other hand, let G be the graph of y = sin(x). As we can see in figure 1.4, the line L hits G in infinitely many points (namely (b,0), where b is any integral multiple of π). Therefore, G cannot be the graph of any polynomial function f (x). So the sine wave is not an algebraic curve. Foranotherexample,let H be the graph of x 2 + y 2 = 1 (a circle). As we can see in figure 1.5, the line L 1 hits the graph H in 2 points, the tangent line L 2 hits H in one point, and the line L 3 hits H in zero points. In a case like this, we take the maximum number of points of intersection, and call it the geometric degree of the curve. So the geometric degree of the circle H is 2. In subsequent chapters, we will discuss how
8 20 CHAPTER 1 L 3 L 2 L 1 Figure 1.5. Three lines intersecting a circle mathematicians dealt with the initially unpleasant but ultimately very productive fact that the number of intersection points is not always the same, but depends on the line we choose as probe. The desire to force this number to be constant, independent of the probing line, turned out to be a fruitful source of new mathematics, as we will see. Similarly to the preceding example, we can look at a sphere, for instance the graph of x 2 + y 2 + z 2 = 1. Again, a line may intersect the sphere in 2, 1, or 0 points. We say the geometric degree of the sphere is 2. To repeat, we provisionally define the geometric degree of a geometric object to be the maximum number of points of intersection of any line with the object. Suppose we have any polynomial f (x, y,...)inany number of variables. It looks very likely from our examples so far that the geometric degree of the graph of f (x, y,...) = 0 should equal the degree of f. 3. Parametric Equations Let s investigate the possibility that the geometric degree of a curve according to the definition of the probing line will equal the degree of the equation of the curve. To do so, we will have to write our probing lines in parametric form. A parameter is an extra variable. A system of equations in parametric form is one where all of the other variables are set equal to functions of the parameter(s). For example, when we describe curves in the xyplane, we will often use t as a parameter, and write x = f (t)and y = g(t). We will only use parametric form with a single parameter. A single parameter will suffice to describe curves, and particularly lines. We use parametric form because that method of describing curves makes it easy
9 DEGREE OF A CURVE 21 to find intersection points of two curves by substituting one equation into another, as you will see. Other reasons to use parametric form will also soon become apparent. Here s an important assumption: We will always parametrize lines linearly. What does this assumption mean? Whenever we parametrize a line, we will always set x = at + b and y = ct + e, where a and c cannot both be 0. It is possible to pick other more complicated ways to parametrize a line, and we want to rule them out. We can describe a line or a curve in the plane in two different ways. We can give an equation for it, such as y = mx + b. Here, m and b are fixed numbers. You probably remember that m is the slope of the line and b is its yintercept. A problem with this form is that a vertical line cannot be described this way, because it has infinite slope. The equation of a vertical line is x = c, for some constant c. We can include all lines in a single formula by using the equation ax + by = c for constants a, b, and c. Depending on what values we assign to a, b, and c, we get the various possible lines. The other way to describe a line or a curve in the plane is to use a parameter. A good way to think about this is to pretend that the line or curve is being described by a moving point. At each moment of time, say at time t, the moving point is at a particular position in the plane, say at the point P(t). We can write down the coordinates of P(t) = (x(t), y(t)). In this way we get two functions of t, namely x(t) and y(t). These two functions describe the line or curve parametrically, where t is the parameter. The line or curve is the set of all the points (x(t), y(t)), as t ranges over a specified set of values. Parametric descriptions are very natural to physicists. They think of the point moving in time, like a planet moving around the sun. The set of all points successively occupied by the moving point is its orbit. For example, the line with equation y = mx + b can be expressed parametrically by the pair of equations x = t and y = mt + b. The parametric description may seem redundant: We used two equations where formerly we needed only one. Each kind of representation has its advantages and disadvantages. One advantage of parametric representation is when we go to higher dimensions. Suppose we have a curve in 6dimensional space. Then we would need five (or perhaps more) equations in six variables to describe
10 22 CHAPTER 1 it. But since a line or curve is intrinsically only 1dimensional, we really should be able to describe it with one independent variable. That s what the parametric representation does for us: We have one variable t and then 6 equations of the form x i = f i (t) for each of the coordinates x 1,..., x 6 in the 6dimensional space in which the curve lies. As we already mentioned, a second advantage of the parametric form is that it gives us a very clear way to investigate the intersection of the line or curve with a geometric object given in terms of equations. Some curves are easy to express in either form. Consider, for example, the curve with equation y 2 = x 3, which can be seen in figure 3.7. This curve can be expressed parametrically with the pair of equations x = t 2, y = t 3. On the other hand, the curve defined by the equation x 4 + 3x 3 y + 17y 2 5xy y 7 = 0 is pretty hard to express parametrically in any explicit way. Conversely, it s hard to find an equation that defines the parametric curve x = t 5 t + 1, y = e t + cos(t) as t runs over all real numbers. For future use, we pause and describe how to parametrize a line in the xyplane. If a, b, c,and e are any numbers, then the pair of equations x = at + b y = ct + e parametrizes a line as long as a or c (or both) are nonzero. Conversely, any line in the xyplane can be parametrized in this way. In particular: If the line is not vertical, it can be described with an equation of the form y = mx + b, and then the pair of equations x = t y = mt + b parametrizes the same line. If the line is vertical, of the form x = e, then thepairofequations gives a parametrization. x = e y = t
11 DEGREE OF A CURVE Our Two Definitions of Degree Clash First, let s look at a simple example to show that the degree of an equation doesn t always equal the geometric degree of the curve it defines. In this section, we only consider real, and not complex, numbers. Let K be the graph of x 2 + y = 0. Because squares of real numbers cannot be negative, K is the empty set. Our definition of the probing line would tell us that K would have geometric degree zero. But the polynomial defining K has degree 2. If you object that we needn t consider empty curves consisting of no points, we can alter this example as follows: Let M be the graph of x 2 + y 2 = 0. Now M consists of a single point, the origin: x = 0, y = 0. By our definition of the probing line, M would have geometric degree 1. But the polynomial defining M has degree 2. You may still object: M isn t a curve it s got only 1 point. But we can beef up this example: Let N be the graph of (x y)(x 2 + y 2 ) = 0. Now N consists of the 45 line, given parametrically by x = t, y = t. Byour definition of the probing line, N would have geometric degree 1. But the polynomial defining N has degree 3. Another type of example would be the curve defined by (x y) 2 = 0. This is again the 45 line, but the degree of the equation is 2, not 1. There is one important observation we can make at this point: If a curve is given by an equation of degree d, any probing line will intersect it in at most d points. Let s see this by looking at our continuing example. Suppose the curve E is given by the equation y 2 = x 3 x. Take the probing line L given by x = at + b, y = ct + e for some real constants a, b, c, and e. As we already mentioned, any line in the plane can be parametrized this way for some choice of a, b, c,and e. When you substitute the values for x and y given by the parametrization into the equation for E, you will get an equation that has to be satisfied by any parameter value for t corresponding to a point of intersection. If we do the substitution, we get (ct + e) 2 = (at + b) 3 (at + b).
12 24 CHAPTER 1 G L L L L Figure 1.6. Four lines and a parabola If we expand using the binomial theorem and regroup terms, we obtain the equation 2 a 3 t 3 + (3a 2 b c 2 )t 2 + (3b 2 a a 2ec)t + b 3 b e = 0 We see that the equation will have degree at most 3, no matter what a, b, c,and e are. Therefore, it will have at most 3 roots. So at most 3 values of t can yield intersection points of E and L. Now this lack of definiteness as to the number of intersection points leads to a serious problem with our probing line definition of degree. Let us suppose we have a curve C and we choose a probing line L and we get 5 points in the intersection of C and L. How do we know 5 is the maximum we can get? Maybe a different line L ' will yield 6 or more points in the intersection of C and L '. How will we know when to stop probing? In fact, we can become greedy. We could hope to redefine the geometric degree of C to be the number of points in the intersection of C and L no matter what line L we pick! This may sound like a tall order, but if we could do it, we d have a beautiful definition of degree. It wouldn t matter what probe we choose. Pick any one and count the intersection points. It may seem like this is hopeless. But let s look at an example. Although it is very simple, this example will show all the problems in our greedy approach to the concept of degree that we will solve in the following three chapters. When we solve them, by redefining the concept of intersection point in an algebraically cogent way, our hope will have come true! Here is the example, which is illustrated in figure 1.6. Let G be the graph of the parabola y = x 2. We consider various different probing lines. For
13 DEGREE OF A CURVE 25 example, let L be the line given parametrically by x = t, y = 3t 2. The intersection of L and G will occur at points on the line with parameter value t exactly when 3t 2 = t 2. This yields the quadratic equation t 2 3t + 2 = 0, which has the two solutions t = 1and t = 2. The two points of the intersection are (1, 3 1 2) = (1, 1) and (2, 3 2 2) = (2, 4). This is the optimal case: We get 2 points of intersection, the most possible for an equation of degree 2. Now look at the probing line L ', given parametrically by x = 0, y = t. This is the vertical line otherwise known as the yaxis. When we plug these values into the equation for the parabola we get t = 0 2, which has only one solution: t = 0, corresponding to the single point of intersection (0, 0). The horizontal probing line L '' given by x = t, y = 0 doesn t fare any better: Plugging in we get 0 = t 2, which again has only one solution: t = 0, corresponding to the single point of intersection (0, 0). Finally, look at the probing line L ''', given parametrically by x = t, y = 5t 10. When we plug these values into the equation for the parabola we get 5t 10 = t 2 or equivalently, t 2 + 5t + 10 = 0. Using the quadratic formula, we see that since the discriminant < 0, there are no solutions for t and hence no points of intersection. Thus, in the simple case of a parabola, we have some probing lines that meet the parabola in 2 points, others that meet the parabola in only 1 point, and others that don t intersect it at all. Yet the equation of a parabola has degree 2. After we finish the next 3 chapters, we will be able to say that any probing line intersects the parabola in 2 points, after we have suitably redefined the concept of intersection. The constructions we will have to make to find a suitable redefinition of intersection will be crucial later for our understanding of elliptic curves and so of the Birch SwinnertonDyer Conjecture.
Algebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationYear 9 set 1 Mathematics notes, to accompany the 9H book.
Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationSOLVING POLYNOMIAL EQUATIONS
C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationLagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.
Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More information13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant
æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More informationAlgebra 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
More informationWhat does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.
PRIMARY CONTENT MODULE Algebra  Linear Equations & Inequalities T37/H37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationSection 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines  specifically, tangent lines.
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationPolynomial Expressions and Equations
Polynomial Expressions and Equations This is a really closeup picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationSECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31
More information9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.
9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n1 x n1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationInteger roots of quadratic and cubic polynomials with integer coefficients
Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationPolynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005
Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division
More informationExamples of Tasks from CCSS Edition Course 3, Unit 5
Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationIf A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationBookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina  Beaufort Lisa S. Yocco, Georgia Southern University
More informationTrigonometric Functions and Equations
Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions  that is, algebraic fractions  and equations which contain them. The reader is encouraged to
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationAlgebra 2 YearataGlance Leander ISD 200708. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 YearataGlance Leander ISD 200708 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationSolving Cubic Polynomials
Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial
More informationChapter 9. Systems of Linear Equations
Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More information3.6. The factor theorem
3.6. The factor theorem Example 1. At the right we have drawn the graph of the polynomial y = x 4 9x 3 + 8x 36x + 16. Your problem is to write the polynomial in factored form. Does the geometry of the
More informationElements of a graph. Click on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and yintercept in the equation of a line Comparing lines on
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationPolynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
More information3.2 The Factor Theorem and The Remainder Theorem
3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More information7.2 Quadratic Equations
476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More information2.5 Zeros of a Polynomial Functions
.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the xaxis and
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationThe PointSlope Form
7. The PointSlope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
More informationDerive 5: The Easiest... Just Got Better!
Liverpool John Moores University, 115 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering
More informationAlgebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationPOLYNOMIALS and FACTORING
POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationMathematics PreTest Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}
Mathematics PreTest Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {1, 1} III. {1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationTim Kerins. Leaving Certificate Honours Maths  Algebra. Tim Kerins. the date
Leaving Certificate Honours Maths  Algebra the date Chapter 1 Algebra This is an important portion of the course. As well as generally accounting for 2 3 questions in examination it is the basis for many
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationPolynomial Operations and Factoring
Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.
More informationAlgebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test
Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationAlex, I will take congruent numbers for one million dollars please
Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohiostate.edu One of the most alluring aspectives of number theory
More information