# REVIEW EXERCISES DAVID J LOWRY

Size: px
Start display at page:

Transcription

1 REVIEW EXERCISES DAVID J LOWRY Contents 1. Introduction 1 2. Elementary Functions Factoring and Solving Quadratics Polynomial Inequalities Rational Functions Exponentials and Logs 7 1. Introduction We have now covered three-fifths of the material from this course. While in many ways, this course has been cumulative and we have revisited much of the earlier material repeatedly, there are things that have been left out. In this packet, we will review some of the problem-types that we have come across and methods of finding their solution. We will also take this time to combine our new skills, such as our knowledge of trigonometric identities and transcendental functions, with some of our old skills. 2. Elementary Functions We started by reviewing basic factoring and graphing linear equations. We then worked on developing the algebraic and graphical qualities of polynomials. We learned to solve any quadratic equation that we will ever see (any cubic too, though that s much harder). Later came the Factor Theorem and the Rational Root Theorem, allowing us to solve more. This all culminated with rational functions. This first set of exercises covers this material Factoring and Solving Quadratics. Example 2.1. One tool that we use with high frequency is factoring. We cannot get away from factoring, it turns out. Many tools revolve around factoring. We ll look at three major factoring tools here. (1) (x + y) 2 = x 2 + 2xy + y 2 and (x y) 2 = x 2 2xy + y 2 Date: August 2,

2 2 DAVID J LOWRY (2) (x y)(x + y) = x 2 y 2 (3) Completing the square Example 2.2. We can use (x + y) 2 = x 2 = 2xy + y 2 and (x y) 2 = x 2 2xy + y 2 both ways. For example, when we see an expression that we d like to expand, like (sin x+cos x) 2, we can immediately say that it is sin 2 x+ cos 2 x + 2 sin x cos x = sin x cos x without any FOILing. (further, if we are exceptionally clever, we might remember that 2 sin x cos x = sin(2x)). On the other had, we can factor functions quickly too. For example, when we are asked to find the roots of e 2x 4e x + 4, we recognize this as (e x 2) 2. Thus there is a double root at x = ln 2. Exercise 2.3. Expand the following without directly multiplying it out: (3x + 2y) 2 (5x 4) 2 (x + cos x) 2 (sec x sin x) 2 (e x + e x ) 2 Exercise 2.4. Factor the following: cos 2 x + 2 cos x tan x + tan x x 2e 2x + 6 2e x + 9 sin 2 x + 8 sin x x 2 + x 2 Example 2.5. It is usually very easy to see cases where we have (x y)(x+y) and rewrite it as x 2 y 2, but we sometimes need to approach it in the opposite direction. For example, if we want to find the roots of sin 2 x 1/2, we can do this quickly and easily with this factoring method. sin 2 x 1/2 = (sin x 1/ 2)(sin x + 1/ 2), and thus the solutions are x = π/4 + nπ/2. Exercise 2.6. Factor the following: 3x 2 2y 2 (just because everything starts as squares doesn t mean that they re the squares of pretty numbers) 4e 2x 9 2 tan x 4 Further, if we want to factor over the complex numbers, we might notice that x 2 + y 2 = (x + iy)(x iy). So we cam factor the following: 4x 2 + 9y 2 2x y 2 Example 2.7. There is a general form for completing the square that always works. If we have x 2 + ax + b, we can note that x 2 + ax + a2 4 a2 4 + b = (x+ a 2 )2 a2 4 +b. For example, x2 +3x+5 = x 2 +3x = (x+ 3 2 ) If we believe this pattern, we could skip the middle. For example, if we had x 2 +10x+3, we could write this as (x ) without any of the middle expansions.

4 4 DAVID J LOWRY this to decide on our inequality. As an aside, I want to mention that this is one of the easiest types of questions to merit partial credit as long as you show your work, if you re in a graded situation. Example Let us solve the quadratic inequality 3x 2 + 4x 1. First, we gather ever everything to one side. So we want to solve 3x 2 + 4x 1 0. Where are the roots of the quadratic 3x 2 + 4x 1? This doesn t immediately seem to factor nicely, so we use the quadratic formula: the roots are x +, x = 4 ± These roots split the real line into the 6 three regions (, x ), (x, x + ), (x +, ). We need to check the sign of our polynomial on each of the three regions. Using, for example, 100, 0, 100, ( we see that the signs go + +. Thus the inequality s solution is x in, 4 ] [ ) 28 and,. 6 6 Exercise Solve the following quadratic inequalities. 8x 2 x > 3 9x 2 + 6x > 1 4x 2 4 The same idea works for higher degree polynomials as well. The task is the same: find the roots, make a number line, identify regions where the polynomial is positive and negative, and use this to find your answer. Also remember - it is not always the case that the sign switches positive negative positive negative. Exercise Solve the following polynomial inequalities. These can be done with factoring. x 2 + 4x 6 6 2x 2 + x 15 < 0 x 2 + 2x x 3 x 2 16x x 3 x 2 16x x 4 x 2 20 > 0 Exercise Solve the following polynomial inequalities. You may have to use other tools, such as the rational root theorem or factor theorem, to proceed here. x 4 x 3 2x 4 > 0 x 5 x 4 3x 3 + 5x 2 2x 2.3. Rational Functions. In many ways, understanding rational functions comes down to understanding polynomials. Once we understand polynomials and, in particular, identifying where they are positive, negative, or zero, we know a tremendous amount about rational functions. The general method of attacking rational functions is to find the zeroes of the numerator and denominator, set up a sign chart with these zeroes as the

5 REVIEW EXERCISES 5 important places, and to identify where the rational function will be positive and where it will be negative. Zeroes of the numerator lead to zeroes of the rational function. Zeroes of the denominator lead to vertical asymptotes of the rational function. If there is the same zero in the numerator and denominator, then there might be a hole. The only bit remaining with respect to rational functions is to understand their limiting behavior. This falls into a few different categories: there might be a horizontal asymptote, a slant asymptote, or no asymptote. Example Consider the rational function f(x) = x2 + 3x + 5 x 3 + x + 1, and suppose we want to find its limiting behavior. If we think of really large x, then x 3 is much larger than x 2. In general, if the degree of the denominator is greater than the degree of the numerator, than the limiting behavior is a horizontal asymptote at y = 0. That is the case here. Example Consider the rational function g(x) = x3 + 3x + 1 4x 3, and + 1 suppose we want to find its limiting behavior. If we think of really large x this time, we can t use the same trick as above. Now the degree of the numerator and denominator are the same. But for really large x, everything except the x 3 and 4x 3 terms matter less and less. In general, if the degrees of the denominator and numerator are the same, then there is a horizontal asymptote. For g(x), we expect g(x) 1 4 for really large x, as the x3 term of the numerator gets divided by 4x 3 in the denominator. This leads to the general fact that the horizontal asymptote in these cases will be at y = a b, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator. Example Consider the rational function h(x) = x3 + 3x + 1 x 2. The + 1 degree of the numerator is exactly one more than the degree of the denominator. Using polynomial long division, we see that h(x) = x + 2x + 1 x 2 + 1, so that for large x the polynomial behaves just like x (the 2x + 1 x as x gets big). We call the line x in this case the slant asymptote, and we find it in general by performing polynomial long division. Example Consider the rational function j(x) = x5 + 3x x 2. Polynomial long division would reveal limiting behavior similar to a cubic, as + 1 the degree of the numerator is 5 and the degree of the denominator is only 2. We don t care about curved asymptotes in this course, so all that we care about here is whether the function goes to or as x and x.

6 6 DAVID J LOWRY Exercise Find the zeroes, vertical asymptotes, holes, horizontal asymptotes, and slant asymptotes of the following rational functions. Sketch the results. (1) x2 5x + 4 x 2 4 (2) 2x 2 5x + 2 4x 2 2x 12 (3) 2x3 x 2 2x + 1 x 2 + 3x + 2 (4) 2x3 + x 2 8x 4 x 2 4x + 2 (similar to, but not the same as, the previous) Exercise Let s see a sort of way in which rational functions might come up. Certain professions, such as any sort of manufacturing or chemical engineer, need to worry about particular types of problems that we call mixing problems. Suppose, for instance, that a large tank contains 50 liters of a 75%/25% water/sodium benzoate solution. We want a larger concentration of sodium benzoate, but it s challenging and expensive to get pure sodium benzoate. But it s easy to get a 75%/25% sodium benzoate/water mixture. So we pour x liters of this new mixture into the tank. Show that the new concentration C (starting at 0.25 and changing because we are adding liquid with a 0.75 concentration) is given by 3x + 50 C = 4(x + 50) Find the limiting behavior of this system (for positive x only - the implied domain of this model is for x positive only. Why is that?). Does this limiting behavior make sense? In fact, mixing problems are very important. But to be fair, this would be one of the easiest mixing problems out there. Chemical engineers, for example, have to work with different concentrations of different materials interacting with each other - and different concentrations change the rate of chemical reaction and interaction as well. There is some intense math there - but this is where it starts. Exercise In my work, I happen to use rational functions quite a bit. There are some miraculous properties of rational functions. For better or worse, we look at two of them here. (1) Often, math asks meta-type questions: instead of what is the solution? it might ask when is this solveable? For example, for what k is the equation x 2 + (1 3k)x + (2 k) = 0 (2.1) solveable for real-valued x? To do this, solve for k. You ll get a rational function. Find the range of that rational function, and this will be the exact vales of k for which equation (2.1) is solveable.

7 REVIEW EXERCISES 7 (2) This introduces a surprising relationship between( rational ) functions, a b matrices, and complex numbers. Given a matrix, we can associate a rational function f(z) = az + b on the complex numbers. c d cz + d ( ) a b There are some stunning things here: if the matrix is invertible with inverse, then the rational function f(z) = az + b g h cz + d c d ( ) e f is invertible with inverse f 1 (z) = ez + f. This is not at all obvious, and is a bit surprising. If you recall the geometry of complex gz + h numbers, and remember that multiplying has to do with a certain rotation and scaling operation, then one can view the associated rational functions to matrices as doing a certain rotation, scaling, shifting, and then doing another roation, scaling, and shifting. The work I do uses this sort of interplay extensively, and this hints at two pervasive concepts of higher mathematics: we find connections between different objects, ultimately learning more about everything involved; and we let things act on other things (in this case, matrices are acting on the complex plane) and through these actions, we learn more about both what s being acted upon and the actor Exponentials and Logs. Exercise Review the basic definitions and properties of exponentials and logarithms. Also review the change-of-base formula. Example Our key interest with exponentials was with modeling certain types of growth. The easiest to remember is compounding interest. If an initial payment of P is put into an account that grows as r percent interest that compounds n times a year, then after t years, there will be P (1 + r n )nt in the account. If the interest compounds continuously, there will be P e rt in the account. Exercise Find the amounting of money in the accounts at the given amount of time shown: (1) placeholder Exercise Let s do an experiment. Suppose you are in college debt, a situation which forces some to get a new loan every 6 months for 4 to 5 years. After some amount of time, you might have to pay back 9 or so different loans, each with their own interest rates. Think to yourself about the following: what s the best way to pay it back? Choose the largest interest account and pay that one off? Distribute money across several accounts? Pay off the interest on each, but focus on one or another? This exercise will be a bit computation heavy, so I recommend that you pull out your calculator,

8 8 DAVID J LOWRY some paper, and keep great notes and a table. It is these notes/table that I ll want to see (1) Let us suppose each of the 9 loans is for 4000 dollars, and they have the following anual interest rates: 3%, 3.5%, 4%, 4.4%, 4.8%, 5%, 5.2%, 5.2%, 5.4%, compounding continuosly (it will give a good approximation). Let us also suppose that we have 500 dollars available per month to pay into these loans, and we invest these at the end of the month, each month. (2) First, let s see what happens if we use a dumb payback scheme: pay off the smallest interest first, and then progress higher. The smallest loan debt would grow like 4000e.03t. After one month, the debt on this account would be 4000e.03(1/12). This is about \$ We then pay in \$500, leaving \$ in the account. The next month, the debt in the account would grow to ( e /03(1/12) (note that we used 1/12 again, as this is the amount of time (in years) that passed from this month to the next. This is about \$ We again pay in our \$500, and we keep on going. We see that on the 9th month, we won t need all \$500. So we use what we need, and then put the next in the next-smallest account. How big is that account now? Looking above, we see it had interest rate 3.5%. After 9 months, it will have grown to size 4000e.035(9/12), or about 4091 dollars. Continue in this fashion, paying off the different debts in this order. How long does it take, and what is the total cost? (3) Now, let s use a better scheme. Pay off the largest interest rates first. How long does it take, and what is the total cost? (4) Now, I give you an option, Either come up with your own payback method to try, or do the following computationally intense method - each month, pay the interest on all accounts, and with the leftovers, pay off the highest account. This isn t actually much harder or longer, once you realize that the interest payment on all but one loan don t change from month to month. (5) Which of these is the best way to pay off one s debt? Note that in every case, there s an interesting property: it s hard to make progress at first, as there is something like \$100 in interest each month. But as you pay more off, the interest rates fall, and it gets easier. This intuition messes with a lot of people s finances. This also leads to the wisdom that large initial payments reduce overall pain by a lot. (6) This is very similar to the financial situation one of my friends found themselves in, except the numbers were not this clean. He got an engineering position, but he worked as a waiter for 2 months as well to supplement his initial payments. Those 2 months ended up reducing the length of his payment period by about 8 months.

### 2 Integrating Both Sides

2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation

PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

### Examples 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

### Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems

Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write

### 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

### Derive 5: The Easiest... Just Got Better!

Liverpool John Moores University, 1-15 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering

### 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

### Chapter 7 Outline Math 236 Spring 2001

Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will

### How To Understand And Solve Algebraic Equations

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

### Some Lecture Notes and In-Class Examples for Pre-Calculus:

Some Lecture Notes and In-Class Examples for Pre-Calculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax

### 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

### Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships

Georgia Standards of Excellence Frameworks Mathematics Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships These materials are for nonprofit educational purposes

### Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 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

### Thnkwell 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

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus 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

### 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

### 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

### MBA Jump Start Program

MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right

### 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.

### 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

### 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

### The Fourth International DERIVE-TI92/89 Conference Liverpool, U.K., 12-15 July 2000. Derive 5: The Easiest... Just Got Better!

The Fourth International DERIVE-TI9/89 Conference Liverpool, U.K., -5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue Notre-Dame Ouest Montréal

### 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

### Graphing Trigonometric Skills

Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE

### Tips for Solving Mathematical Problems

Tips for Solving Mathematical Problems Don Byrd Revised late April 2011 The tips below are based primarily on my experience teaching precalculus to high-school students, and to a lesser extent on my other

### What 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

### 1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives

TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### 3.2 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

### 0.8 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

### SECTION 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

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### A 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

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### Algebra 2: Themes for the Big Final Exam

Algebra : Themes for the Big Final Exam Final will cover the whole year, focusing on the big main ideas. Graphing: Overall: x and y intercepts, fct vs relation, fct vs inverse, x, y and origin symmetries,

### 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

### 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

### 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

### Clovis Community College Core Competencies Assessment 2014 2015 Area II: Mathematics Algebra

Core Assessment 2014 2015 Area II: Mathematics Algebra Class: Math 110 College Algebra Faculty: Erin Akhtar (Learning Outcomes Being Measured) 1. Students will construct and analyze graphs and/or data

### ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals

ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an

### 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

### 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

### WARM UP EXERCSE. 2-1 Polynomials and Rational Functions

WARM UP EXERCSE Roots, zeros, and x-intercepts. x 2! 25 x 2 + 25 x 3! 25x polynomial, f (a) = 0! (x - a)g(x) 1 2-1 Polynomials and Rational Functions Students will learn about: Polynomial functions Behavior

### 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:

### Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

### MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial

### 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

### Factoring Trinomials: The ac Method

6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

### BookTOC.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

### NSM100 Introduction to Algebra Chapter 5 Notes Factoring

Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

### 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

### 6.4 Logarithmic Equations and Inequalities

6.4 Logarithmic Equations and Inequalities 459 6.4 Logarithmic Equations and Inequalities In Section 6.3 we solved equations and inequalities involving exponential functions using one of two basic strategies.

### 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

### Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

### 9. 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 n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

### Functions and their Graphs

Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers

### correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

### Administrative - Master Syllabus COVER SHEET

Administrative - Master Syllabus COVER SHEET Purpose: It is the intention of this to provide a general description of the course, outline the required elements of the course and to lay the foundation for

### Algebra 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

### 3.6. 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

### 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

### March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### Study Guide 2 Solutions MATH 111

Study Guide 2 Solutions MATH 111 Having read through the sample test, I wanted to warn everyone, that I might consider asking questions involving inequalities, the absolute value function (as in the suggested

### a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F

FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all

### Differentiation and Integration

This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

### SECTION 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

### 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

### Graphic Designing with Transformed Functions

Math Objectives Students will be able to identify a restricted domain interval and use function translations and dilations to choose and position a portion of the graph accurately in the plane to match

### Algebra 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

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

### 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.

### Factoring Polynomials

Factoring Polynomials Any Any Any natural number that that that greater greater than than than 1 1can can 1 be can be be factored into into into a a a product of of of prime prime numbers. For For For

### http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304

MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio

### 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,

### Year 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

### Prerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.

Course Syllabus Math 1314 College Algebra Revision Date: 8-21-15 Catalog Description: In-depth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems

### 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

### ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

### Polynomial Expressions and Equations

Polynomial Expressions and Equations This is a really close-up picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen

### 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

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

### MATH 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.

### Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

### 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

### Algebra II EOC Practice Test

Algebra II EOC Practice Test Name Date 1. Suppose point A is on the unit circle shown above. What is the value of sin? (A) 0.736 (B) 0.677 (C) (D) (E) none of these 2. Convert to radians. (A) (B) (C) (D)

### Graphing Rational Functions

Graphing Rational Functions A rational function is defined here as a function that is equal to a ratio of two polynomials p(x)/q(x) such that the degree of q(x) is at least 1. Examples: is a rational function

### Math 131 College Algebra Fall 2015

Math 131 College Algebra Fall 2015 Instructor's Name: Office Location: Office Hours: Office Phone: E-mail: Course Description This course has a minimal review of algebraic skills followed by a study of

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Chapter 7. Functions and onto. 7.1 Functions

Chapter 7 Functions and onto This chapter covers functions, including function composition and what it means for a function to be onto. In the process, we ll see what happens when two dissimilar quantifiers

### INTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y).

INTERPOLATION Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). As an example, consider defining and x 0 =0, x 1 = π 4, x

### Math 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

### FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x

### TRIGONOMETRY Compound & Double angle formulae

TRIGONOMETRY Compound & Double angle formulae In order to master this section you must first learn the formulae, even though they will be given to you on the matric formula sheet. We call these formulae

### CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises

CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =

### MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:

MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an

### Zeros of Polynomial Functions

Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate