The Number of Real Roots of a Cubic Equation

Size: px
Start display at page:

Download "The Number of Real Roots of a Cubic Equation"

Transcription

1 The Number of Real Roots of a Cubic Equation Richard Kavinoky Santa Rosa Junior College J. B. Thoo Yuba College November 6, 2007 The number of distinct real roots 1 of the cubic equation x + bx 2 + cx + d = 0 (1) equals, for example, the number of normals to the parabola y = x 2 through a given point in the plane (Bains and Thoo (2007)), 2 as well as the number of equilibrium solutions of dx/dt = x + bx 2 + cx + d. Now, one way to find the number of real roots of (1) is to solve the equation. Certainly, (1) can be solved by hand easily if d = 0. If d 0, the equation can still be solved by hand using Cardan s (or Cardano s) cubic formula (Fine (1961); Gellert et al. (1975)), but not easily in general. Of course, using computer software or a graphing calculator can make light work of solving (1) altogether. However, there is a certain satisfaction in being able to tell the number of real roots of (1) without first solving the equation. It turns out that this is easy to do. The key is the discriminant. Every intermediate algebra student learns the quadratic formula, deriving it by completing the square. And with the quadratic formula in hand, it is apparent that the number of real roots of the quadratic equation x 2 +bx+c = Mendocino Avenue, Santa Rosa, CA , rkavinoky@santarosa.edu N. Beale Road, Marysville, CA , jthoo@yccd.edu. 1 We mean the number of distinct real roots throughout. 2 It is remarkable that Apollonius had obtained precise results on the number of normals to a parabola through a given point using purely synthetic geometry (Heath, 1981, pp , ) almost 1700 years before Italian mathematicians solved the cubic. For Cardan s formulation of the solution of the cubic, as well as the story behind the quarrel between Cardan and Tartaglia over its publication, see Burton (200) for example. For other formulations of the solution of the cubic, see Kalman and White (1998) and the references therein. 1

2 is determined by its discriminant, b 2 4c. But very few students today see Cardan s cubic formula, and its derivation is much less straightforward than that of the quadratic formula. So, how may a student today come up with or be led to the discriminant of the cubic equation (1) without appealing to the cubic formula? In this note, we present one way of doing so using ideas from a first calculus course derivative, critical point, local extrema, and graphing in an intuitive way. We also show how the discriminant defines a boundary in the plane across which the number of real roots of (1) changes, and apply the discriminant to determining the number of normals to the parabola y = x 2 through a given point and the number of equilibrium solutions of dx/dt = (R R c )x ax, the Landau equation in fluid mechanics (Boyce and DiPrima, 200, p. 89), where R c and a are positive constants and R is a parameter. Discriminant of a cubic equation A good way to begin is by looking at the graph of y = x + bx 2 + cx + d. To simplify the analysis, we translate the graph horizontally through the change of variables x x b/, thereby obtaining the graph of y = x + px + q that has its inflection point on the y axis, and examine equivalently the number of real roots of the reduced cubic equation x + px + q = 0. (2) Toward this end, we define the function C(x) = x + px + q and consider the following cases. I: p 0 x + px + q = 0 II: p < 0 A: q = 0 B: q < 0 C: q > 0 Case I: p 0 2

3 If p = 0, then C(x) = x + q, so (2) has one real root, namely, q 1/. Otherwise, C (x) 0 for all x, so C is monotone increasing and, hence, (2) again has one real root. Case II.A: p < 0 and q = 0 Then C(x) = x + px, so (2) has three distinct real roots: ( p) 1/2, 0, and ( p) 1/2. For the next two cases, we note that the critical points of C are ) 1/2 x = and x + = ) 1/2, and that the second derivative test implies that C(x ) is a local maximum and C(x + ) is a local minimum. Case II.B: p < 0 and q < 0 Then C(0) < 0. This gives three possibilities for the graph of y = C(x); see Figure 1. It follows that (2) has one real root if and only if C(x ) < 0; two distinct real roots if and only if C(x ) = 0; and three distinct real roots if and only if C(x ) > 0. x x + x x + x x + (a) C(x ) < 0 (b) C(x ) = 0 (c) C(x ) > 0 Figure 1: Case II.B: p < 0 and q < 0. First suppose that C(x ) < 0. Using p we find that C(x ) = ) 1/2 = ( p) 2/2 ) /2 p ) 1/2 ) /2, = ) 1/2 ) /2 + q = 2 + q.

4 Thus, C(x ) < 0 if and only if q ) /2. 2 < Since both sides of this are negative, squaring yields the equivalent inequality q 2 ), 4 > which is equivalent to 4 + p 27 > 0. Therefore, (2) has one real root if and only if q 2 /4 + p /27 > 0. q 2 Next suppose that C(x ) = 0 or C(x ) > 0. A similar analysis shows that (2) has two distinct real roots if an only if q 2 /4 + p /27 = 0, and three distinct real roots if and only if q 2 /4 + p /27 < 0. Case II.C: p < 0 and q > 0 Then C(0) > 0. This again gives three possibilities for the graph of y = C(x); see Figure 2. Now it follows that (2) has one real root if and only if C(x + ) > 0; two distinct real roots if and only if C(x + ) = 0; and three distinct real roots if and only if C(x + ) < 0. x x + x x + x x + (a) C(x + ) > 0 (b) C(x + ) = 0 (c) C(x + ) < 0 Figure 2: Case II.C: p < 0 and q > 0. Following the analysis of Case II.B, we find that (2) has one real root if and only if q 2 /4 + p /27 > 0; two distinct real roots if and only if q 2 /4 + p /27 = 0; and three distinct real roots if and only if q 2 /4 + p /27 < 0. What we have shown, therefore, is that the number of real roots of the cubic equation (1) is characterized by D = q2 4 + p 27, 4

5 called the discriminant of the equation, where p = c b2 and q = 2b 27 bc + d. We summarize this in the following proposition. Proposition 1 If p = q = 0, then the cubic equation x + bx 2 + cx + d = 0 has one root, namely, zero. Otherwise, the cubic equation has one real root if and only if D > 0. two distinct real roots if and only if D = 0. three distinct real roots if and only if D < 0. We remark that the discriminant D = q 2 /4 + p /27 appears in Cardan s cubic formula. Geometry of the discriminant The discriminant q 2 /4 + p /27 gives some insight into the number of real roots of (1). Specifically, the graph of q 2 /4+p /27 = 0 is a boundary in the qp plane across which the number of real roots of (1) changes in the same way that the graph of b 2 4c = 0 is a boundary in the bc plane across which the number of real roots of a quadratic equation x 2 + bx + c = 0 changes. See Figure. Two applications We apply the discriminant D = q 2 /4 + p /27 to determining the number of normals to the parabola y = x 2 through a given point and the number of equilibrium solutions of the Landau equation dx/dt = (R R c )x ax. A normal to the parabola y = x 2 through the point (α, β) satisfies the relation (y β)/(x α) = 1/(2x) that simplifies to the reduced cubic equation (2) with p = 1 2β 2 and q = α 2. Thus, given (α, β), Proposition 1 tells us precisely the number of normals to the parabola y = x 2 through the point. Better yet, if we set D = 0 and use the 5

6 10 p 5 D > 0: one real root 10 c 5 D < 0: no real root D < 0: three real roots 5 D > 0: two real roots q 10 (a) Cubic equation: D = q 2 /4 + p / b 10 (b) Quadratic equation: D = b 2 4c Figure : The graph of the discriminant D = 0 is a boundary across which the number of real roots changes. above relations for p and q with α = x and β = y, we obtain the function y = 2 4/ x2/ = N(x). Figure 4 shows the graph of y = N(x), a semicubical parabola just like the boundary curve in Figure (a), together with the parabola y = x 2. The semicubical parabola is often called Neil s (or Neile s) parabola (Bains and Thoo (2007); Gellert et al. (1975)). Geometrically, then, there is one normal to the parabola y = x 2 through the point (α, β) if the point lies below or on the cusp of Neil s parabola y = N(x); two normals if the point lies on Neil s parabola, except on the cusp; and three normals if the point lies above Neil s parabola. Turning to the Landau equation dx dt = (R R c)x ax, () where R c and a are positive constants and R is a parameter, an equilibrium solution is one such that dx/dt = 0. Thus, an equilibrium solution of () is a solution of the reduced cubic equation (2) with p = R c R a and q = 0. Proposition 1 then implies that () has one equilibrium solution if R R c and three equilibrium solutions if R > R c. Further, the proposition implies that () cannot have two equilibrium solutions. Geometrically, the point (0, (R c R)/a) 6

7 Figure 4: Neil s parabola (heavy curve) together with the parabola y = x 2. lies above or on the cusp of the boundary curve shown in Figure (a) if R R c, and (0, (R c R)/a) lies below the curve if R > R c. The value of the parameter R = R c is called a bifurcation point. Note that here the bifurcation point corresponds to (0, 0), the cusp of the boundary curve. More generally, if we modify the Landau equation () by adding a nonzero constant b to the right-hand side, dx dt = b + (R R c)x ax, then Proposition 1 implies that the equation has one equilibrium solution if R < R c + ab 2 /4 and three equilibrium solutions if R > R c + ab 2 /4. But, unlike (), the modified equation can also have two equilibrium solutions. This occurs if R = R c + ab 2 /4. Note that, because b 0, the point ( b/a, (R c R)/a) does not pass through the cusp of the boundary curve in Figure (a) as the parameter R varies. References Bains, M. S. and Thoo, J. B. (2007). The normals to a parabola and the real roots of a cubic. Coll. Math. J., 8(4): Boyce, W. E. and DiPrima, R. C. (200). Elementary Differential Equations and Boundary Value Problems. John Wiley & Sons, Inc., New Jersey, 7th edition. 7

8 Burton, D. M. (200). The History of Mathematics: An Introduction. McGraw- Hill, Boston, 5th edition. Fine, H. B. (1961). College Algebra. Dover Publications, Inc., New York. Gellert, W., Kustner, H., Hellwich, M., and Kastner, H., editors (1975). The VNR Concise Encyclopedia of Mathematics. Van Norstrand Reinhold Co., New York. Heath, T. (1981). A History of Greek Mathematics Volume II: From Aristarchus to Diophantus. Dover Publications, Inc., New York. Kalman, D. and White, J. (1998). A simple solution of the cubic. Coll. Math. J., 29(5):

PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN 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 information

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

3.3. 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 information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST 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 information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 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 information

Week 1: Functions and Equations

Week 1: Functions and Equations Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter

More information

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

More information

QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS

QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS Content 1. Parabolas... 1 1.1. Top of a parabola... 2 1.2. Orientation of a parabola... 2 1.3. Intercept of a parabola... 3 1.4. Roots (or zeros) of a parabola...

More information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 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 information

Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Copyrighted Material. Chapter 1 DEGREE OF A CURVE 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

More information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

The 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 information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

1 Mathematical Models of Cost, Revenue and Profit

1 Mathematical Models of Cost, Revenue and Profit Section 1.: Mathematical Modeling Math 14 Business Mathematics II Minh Kha Goals: to understand what a mathematical model is, and some of its examples in business. Definition 0.1. Mathematical Modeling

More information

Equations, Inequalities & Partial Fractions

Equations, 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 information

Factoring - Solve by Factoring

Factoring - Solve by Factoring 6.7 Factoring - Solve by Factoring Objective: Solve quadratic equation by factoring and using the zero product rule. When solving linear equations such as 2x 5 = 21 we can solve for the variable directly

More information

Solving Quadratic Equations

Solving 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 information

Solving Cubic Polynomials

Solving 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 information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 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 information

2013 MBA Jump Start Program

2013 MBA Jump Start Program 2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of

More information

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

x 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 information

QUADRATIC EQUATIONS AND FUNCTIONS

QUADRATIC EQUATIONS AND FUNCTIONS Douglas College Learning Centre QUADRATIC EQUATIONS AND FUNCTIONS Quadratic equations and functions are very important in Business Math. Questions related to quadratic equations and functions cover a wide

More information

Factoring Polynomials and Solving Quadratic Equations

Factoring Polynomials and Solving Quadratic Equations Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3

More information

On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems

On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems Dynamics at the Horsetooth Volume 2, 2010. On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems Eric Hanson Department of Mathematics Colorado State University

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

1 Lecture: Integration of rational functions by decomposition

1 Lecture: Integration of rational functions by decomposition Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

More information

Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS

Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS DUSP 11.203 Frank Levy Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding

More information

Assessment Schedule 2013

Assessment Schedule 2013 NCEA Level Mathematics (9161) 013 page 1 of 5 Assessment Schedule 013 Mathematics with Statistics: Apply algebraic methods in solving problems (9161) Evidence Statement ONE Expected Coverage Merit Excellence

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING 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 information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope 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 information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

Understanding Basic Calculus

Understanding 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 information

CURVE FITTING LEAST SQUARES APPROXIMATION

CURVE FITTING LEAST SQUARES APPROXIMATION CURVE FITTING LEAST SQUARES APPROXIMATION Data analysis and curve fitting: Imagine that we are studying a physical system involving two quantities: x and y Also suppose that we expect a linear relationship

More information

Solving for the Roots of the Cubic Equation. Finding the solution to the roots of a polynomial equation has been a fundamental

Solving for the Roots of the Cubic Equation. Finding the solution to the roots of a polynomial equation has been a fundamental Dallas Gosselin and Jonathan Fernandez Professor Buckmire April 18, 014 Complex Analysis Project Solving for the Roots of the Cubic Equation Finding the solution to the roots of a polynomial equation has

More information

1.3 Algebraic Expressions

1.3 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 information

Section 3.1 Quadratic Functions and Models

Section 3.1 Quadratic Functions and Models Section 3.1 Quadratic Functions and Models DEFINITION: A quadratic function is a function f of the form fx) = ax 2 +bx+c where a,b, and c are real numbers and a 0. Graphing Quadratic Functions Using the

More information

Partial Fractions Examples

Partial Fractions Examples Partial Fractions Examples Partial fractions is the name given to a technique of integration that may be used to integrate any ratio of polynomials. A ratio of polynomials is called a rational function.

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 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 information

Zeros of a Polynomial Function

Zeros 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 information

Algebra 2: Q1 & Q2 Review

Algebra 2: Q1 & Q2 Review Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short

More information

Integer roots of quadratic and cubic polynomials with integer coefficients

Integer 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 information

Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Algebra 2 Year-at-a-Glance Leander ISD 2007-08 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 information

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

More information

the points are called control points approximating curve

the points are called control points approximating curve Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

Solutions to Homework 10

Solutions to Homework 10 Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

More information

PARABOLAS AND THEIR FEATURES

PARABOLAS AND THEIR FEATURES STANDARD FORM PARABOLAS AND THEIR FEATURES If a! 0, the equation y = ax 2 + bx + c is the standard form of a quadratic function and its graph is a parabola. If a > 0, the parabola opens upward and the

More information

4.3 Lagrange Approximation

4.3 Lagrange Approximation 206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

More information

Compute the derivative by definition: The four step procedure

Compute the derivative by definition: The four step procedure Compute te derivative by definition: Te four step procedure Given a function f(x), te definition of f (x), te derivative of f(x), is lim 0 f(x + ) f(x), provided te limit exists Te derivative function

More information

Tim Kerins. Leaving Certificate Honours Maths - Algebra. Tim Kerins. the date

Tim 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 information

3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

More information

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary) Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

More information

A Different Way to Solve Quadratics Bluma s Method*

A Different Way to Solve Quadratics Bluma s Method* A Different Way to Solve Quadratics Bluma s Method* Abstract In this article, we introduce an approach to finding the solutions of a quadratic equation and provide two proofs of its correctness. This method

More information

Covariance and Correlation

Covariance and Correlation Covariance and Correlation ( c Robert J. Serfling Not for reproduction or distribution) We have seen how to summarize a data-based relative frequency distribution by measures of location and spread, such

More information

5. Factoring by the QF method

5. Factoring by the QF method 5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the

More information

1 Shapes of Cubic Functions

1 Shapes of Cubic Functions MA 1165 - Lecture 05 1 1/26/09 1 Shapes of Cubic Functions A cubic function (a.k.a. a third-degree polynomial function) is one that can be written in the form f(x) = ax 3 + bx 2 + cx + d. (1) Quadratic

More information

More Quadratic Equations

More Quadratic Equations More Quadratic Equations Math 99 N1 Chapter 8 1 Quadratic Equations We won t discuss quadratic inequalities. Quadratic equations are equations where the unknown appears raised to second power, and, possibly

More information

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

More information

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x). .7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational

More information

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper. FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

More information

BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES.

BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES. BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES. I. GENERALITIES There are 3 common methods to solve quadratic inequalities. Therefore, students sometimes are confused to select the fastest and the best

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

LAB 11: MATRICES, SYSTEMS OF EQUATIONS and POLYNOMIAL MODELING

LAB 11: MATRICES, SYSTEMS OF EQUATIONS and POLYNOMIAL MODELING LAB 11: MATRICS, SYSTMS OF QUATIONS and POLYNOMIAL MODLING Objectives: 1. Solve systems of linear equations using augmented matrices. 2. Solve systems of linear equations using matrix equations and inverse

More information

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of

More information

MATH 21. College Algebra 1 Lecture Notes

MATH 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 information

GRE Prep: Precalculus

GRE Prep: Precalculus GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach

More information

Section 1.1. Introduction to R n

Section 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 information

1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.

1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved. 1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs

More information

CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation

CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of

More information

Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %

Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 % Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the

More information

Answer Key for California State Standards: Algebra I

Answer 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 information

FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

More information

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 y = mx + b.

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 y = mx + b. PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 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 information

Higher Education Math Placement

Higher 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 information

MATH 52: MATLAB HOMEWORK 2

MATH 52: MATLAB HOMEWORK 2 MATH 52: MATLAB HOMEWORK 2. omplex Numbers The prevalence of the complex numbers throughout the scientific world today belies their long and rocky history. Much like the negative numbers, complex numbers

More information

Exam 1 Sample Question SOLUTIONS. y = 2x

Exam 1 Sample Question SOLUTIONS. y = 2x Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

More information

National 5 Mathematics Course Assessment Specification (C747 75)

National 5 Mathematics Course Assessment Specification (C747 75) National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for

More information

Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 51 First Exam January 29, 2015 Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

More information

Warm-Up Oct. 22. Daily Agenda:

Warm-Up Oct. 22. Daily Agenda: Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment

More information

SOLVING QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD

SOLVING QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD SOLVING QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD A quadratic equation in one variable has as standard form: ax^2 + bx + c = 0. Solving it means finding the values of x that make the equation true.

More information

MODERN APPLICATIONS OF PYTHAGORAS S THEOREM

MODERN APPLICATIONS OF PYTHAGORAS S THEOREM UNIT SIX MODERN APPLICATIONS OF PYTHAGORAS S THEOREM Coordinate Systems 124 Distance Formula 127 Midpoint Formula 131 SUMMARY 134 Exercises 135 UNIT SIX: 124 COORDINATE GEOMETRY Geometry, as presented

More information

Solve Quadratic Equations by the Quadratic Formula. The solutions of the quadratic equation ax 2 1 bx 1 c 5 0 are. Standardized Test Practice

Solve Quadratic Equations by the Quadratic Formula. The solutions of the quadratic equation ax 2 1 bx 1 c 5 0 are. Standardized Test Practice 10.6 Solve Quadratic Equations by the Quadratic Formula Before You solved quadratic equations by completing the square. Now You will solve quadratic equations using the quadratic formula. Why? So you can

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

THREE DIMENSIONAL GEOMETRY

THREE DIMENSIONAL GEOMETRY Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

Polynomial Operations and Factoring

Polynomial 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 information

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

More information

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

More information

Algebra II. Weeks 1-3 TEKS

Algebra II. Weeks 1-3 TEKS Algebra II Pacing Guide Weeks 1-3: Equations and Inequalities: Solve Linear Equations, Solve Linear Inequalities, Solve Absolute Value Equations and Inequalities. Weeks 4-6: Linear Equations and Functions:

More information

Integrals of Rational Functions

Integrals of Rational Functions Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t

More information

Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

More information

6.1 Add & Subtract Polynomial Expression & Functions

6.1 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 information

On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples

On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a

More information

3.6 The Real Zeros of a Polynomial Function

3.6 The Real Zeros of a Polynomial Function SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,

More information

Mathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)

Mathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI) ( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =

More information

Algebra 2/Trig Unit 2 Notes Packet Period: Quadratic Equations

Algebra 2/Trig Unit 2 Notes Packet Period: Quadratic Equations Algebra 2/Trig Unit 2 Notes Packet Name: Date: Period: # Quadratic Equations (1) Page 253 #4 6 **Check on Graphing Calculator (GC)** (2) Page 253 254 #20, 26, 32**Check on GC** (3) Page 253 254 #10 12,

More information

3. INNER PRODUCT SPACES

3. INNER PRODUCT SPACES . INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

More information

Don't Forget the Differential Equations: Finishing 2005 BC4

Don't Forget the Differential Equations: Finishing 2005 BC4 connect to college success Don't Forget the Differential Equations: Finishing 005 BC4 Steve Greenfield available on apcentral.collegeboard.com connect to college success www.collegeboard.com The College

More information

Mathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}

Mathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11} Mathematics Pre-Test 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 information

UNCORRECTED PAGE PROOFS

UNCORRECTED PAGE PROOFS number and and algebra TopIC 17 Polynomials 17.1 Overview Why learn this? Just as number is learned in stages, so too are graphs. You have been building your knowledge of graphs and functions over time.

More information

Eigenvalues, Eigenvectors, and Differential Equations

Eigenvalues, Eigenvectors, and Differential Equations Eigenvalues, Eigenvectors, and Differential Equations William Cherry April 009 (with a typo correction in November 05) The concepts of eigenvalue and eigenvector occur throughout advanced mathematics They

More information

Intermediate Value Theorem, Rolle s Theorem and Mean Value Theorem

Intermediate Value Theorem, Rolle s Theorem and Mean Value Theorem Intermediate Value Theorem, Rolle s Theorem and Mean Value Theorem February 21, 214 In many problems, you are asked to show that something exists, but are not required to give a specific example or formula

More information