# Rational inequality. Sunil Kumar Singh. 1 Sign scheme or diagram for rational function

Save this PDF as:

Size: px
Start display at page:

Download "Rational inequality. Sunil Kumar Singh. 1 Sign scheme or diagram for rational function"

## Transcription

2 OpenStax-CNX module: m Let us consider an example here : f (x) = x2 x 2 (x + 1) (x 2) x 2 = 3x 8 (x + 1) (x 4) Critical points are -1, 2, -1 and 4. There are two important things to realize here. First, we can not cancel common linear factors as this will result in loosing undened points and will loose information on sign change. The marking on real number line is as shown here : Figure 1: Second, the fact that function may change its sign in the domain has an interesting consequence. It can be better understood in terms of function graph, which is essentially a curve. The event of crossing of x-axis by the graph records the event of change of sign. Another change in the sign of graph warrants that curve should cross x-axis again. This corresponds to reversal of sign. It is not possible to change sign of function without crossing x-axis. This means that function will change sign at critical points. Equivalently, we say that sign of function alternates in consecutive sub-intervals. Now, these considerations set up the rst two steps of sign diagram : 1: Decompose both numerator and denominator into linear factors. Find critical points by equating linear factors individually to zero. 2: Mark critical points on a real number line. If n be the numbers of critical points, then real number line is divided into (n+1) sub-intervals. The question however remains that we should know sign of function in at least one interval. We determine the same by testing function value for an intermediate x-value in any of the sub-intervals. Though it is not a rule, we consider a test point in the right most interval, which extends to positive innity. This helps us to assign signs in the intervals left to it by alternating signs. Sometimes, it may, however, be easier to evaluate function value at x= 0, 1 or -1, provided they are not the critical points. This has the advantage that calculation of function value is easier. Now, these consideration set up the next step of sign diagram : 3: Test sign of function in a particular interval. Assign alternate signs in adjacent sub-intervals. For the example case, let us put x=0, f (0) = = 1 4 > 0 Thus, sign of function in the interval between -1 and 2 is positive. The signs of function alternate in adjacent sub-intervals.

3 OpenStax-CNX module: m Figure 2: We have noted that sign of each linear factor combines to determine the sign of rational function. This fact is reected as sign alternates in adjacent sub-intervals. However, we need to consider the eect of case in which a linear factor is repeated. If a linear factor evaluates to a positive number in an interval and is repeated, then there is no eect on the sign of function. If a linear factor evaluates to a negative number in an interval and is repeated even times, then there is no eect on the sign of function. The product of negative sign repeated even times yield a positive sign and as such does not aect the sign of function. However, if a linear factor evaluates to a negative number in an interval and is repeated odd times, then sign of function changes. Product of negative sign repeated odd times yield a negative sign and as such sign of function changes. We conclude that if a linear factor is repeated even times, then sign of function will not alternate about the critical point corresponding to linear factor in question. On the other hand, if a linear factor is repeated odd times, then sign of function will alternate as before. Now, these consideration set up the next step of sign diagram : 4: If a linear factor is repeated even times, then sign of function will not alternate about the critical point corresponding to linear factor in question. In the example case, the linear factor (x+1) is repeated even times (count both in numerator and denominator). As such, sign of function will not change about critical point -1. Thus, sign diagram drawn as above need to be modied as : Figure 3: We can verify modication due to repeated linear factors by putting x = -2 in the function : f ( 2) = ( 2)2 ( 2) 2 ( 2) 2 3 ( 2) 8 = = 5 2 > 0 We summarize steps for drawing sign scheme/ diagram as :

4 OpenStax-CNX module: m : Decompose both numerator and denominator into linear factors. Do not cancel common linear factors. Find critical points by equating linear factors individually to zero. 2: Mark distinct critical points on a real number line. If n be the numbers of distinct critical points, then real number line is divided into (n+1) sub-intervals. 3: Test sign of function in a particular interval. Assign alternate signs in adjacent sub-intervals. 4: If a linear factor is repeated even times, then sign of function will not alternate about the critical point corresponding to linear factor in question. 2 Solution of rational inequalities using sign scheme or diagram An important point about interpreting sign diagram is that sign of function relates to non-zero values of function. Note that zero does not have sign. The critical points corresponding to numerator function are zeroes of rational function. As such, the graph of function is continuous at these critical points and these critical points can be included in the sub-interval. On the other hand, the rational function is not dened for critical points corresponding to denominator function (as denominator turns zero). We, therefore, conclude that an interval can include critical points corresponding to numerator function, but not the critical points corresponding to denominator function. In case, there are common critical points between numerator and denominator, then those critical points can not be included in the sub-interval. We can interpret sign diagram in two ways. Either we determine the solution of a given quadratic inequality or we determine intervals of all four types of inequalities for a given quadratic expression. We shall illustrate these two approaches by working with the example case. 2.1 Determining solution of a given quadratic inequality Let us consider that we are required to solve rational inequality f (x) = x2 x 2 x 2 3x 8 0 The sign diagram as drawn earlier for the given rational function is shown here : Figure 4: We need to interpret signs of dierent intervals to nd the solution of a given rational inequality. Clearly, solution of given inequality is : x (, 2] U (4, ) { 1, 4} Note that we need to remove -1 and 4 from the solution set as function is not dened for this x value. However, inequality involved greater than or equal to is not strict inequality. It allows equality to zero. As such, we include critical point 2 belonging to numerator function. Further, we can also write the solution set in alternate form as :

5 OpenStax-CNX module: m x (, 1) U ( 1, 2] U (4, ) 2.2 Determining interval of four quadratic inequalities Let us take the rational expression of example case and determine intervals of each of four inequalities. The sign diagram as drawn earlier is shown here : Figure 5: f (x) < 0; x (2, 4) f (x) 0; x [2, 4) f (x) > 0; x (, 1) U ( 1, 2) U (4, ) f (x) 0; x (, 1) U ( 1, 2] U (4, ) Note that critical point 2 belonging to numerator is included for inequalities which allows equality. Inclusion and exclusion of critical points Based on the discussion above, we summarize inclusion or exclusion of critical points here : 1: Question of inclusion of critical points arises when inequality involved is not strict. 2: Critical points belonging to numerator are included in solution set. 3: Critical points belonging to denominator are excluded from solution set. 3: Critical points belonging to both numerator and denominator are excluded from solution set. 3 Solution of rational inequalities using wavy curve method Wavy curve method is a modied sign diagram method. This method has the advantage that we do not need to test sign of interval as required in earlier case. The steps involved are : 1: Factorize numerator and denominator into linear factors. 2: Make coecients of x positive in all linear factors. This step may require to change sign of x in the linear factor by multiplying inequality with -1. Note that this multiplication will change the inequality sign as well. For example, less than will become greater than etc. 3: Equate each linear factor to zero and nd values of x in each case. The values are called critical points. 4: Identify distinct critical points on real number line. The n numbers of distinct critical points divide real number lines in (n+1) sub-intervals.

6 OpenStax-CNX module: m : The sign of rational function in the right most interval is positive. Alternate sign in adjoining intervals on the left. 5: If a linear factor is repeated even times, then sign of function will not alternate about the critical point corresponding to linear factor in question. We need to exclude exception points i.e. critical points of denominator from solution set. Further, it is important to understand that signs of intervals as determined using this method are not the signs of function rather signs of modied function in which sign of x has changed. However, if we are not required to change the sign of x i.e. to modify the function, then signs of intervals are also signs of function. We shall though keep this dierence in mind, but we shall refer signs of intervals as sign scheme or diagram in this case also. Example 1 Problem : Apply wavy curve method to nd the interval of x for the inequality given : x 1 x 0 Solution : We change the sign of "x" in the denominator to positive by multiplying both sides of inequality with -1. Note that this changes the inequality sign as well. Here, critical points are : x x 1 0 x = 0, 1 The critical points are marked on the real number line. Starting with positive sign in the right most interval, we denote signs of adjacent intervals by alternating sign. Sign diagram Figure 6: Sign of function alternates. Thus, interval of x as solution of inequality is : 0 x < 1 We do not include "1" as it reduces denominator to zero. Example 2 Problem : Find solution of the rational inequality given by : 3x 2 + 6x 15 (2x 1) (x + 3) 1 Solution : We rst convert the given inequality to standard form f(x) 0.

7 OpenStax-CNX module: m x2 + 6x 15 (2x 1) (x + 3) 1 0 3x2 + 6x 15 (2x 1) (x + 3) (2x 1) (x + 3) 3x2 + 6x 15 ( 2x 2 + 5x 3 ) (2x 1) (x + 3) x2 + x 12 (2x 1) (x + 3) 0 x2 + 4x 3x 12 (2x 1) (x + 3) 0 (x 3) (x + 4) (2x 1) (x + 3) 0 Critical points are -4, -3, 1/2, 3. Corresponding sign diagram is : Sign diagram 0 0 Figure 7: Sign of function alternates. The solution of inequality is : x (, 4] ( 3, 1/2) [3, ) We do not include "-3" and "1" as they reduce denominator to zero. Example 3 Problem : Find solution of : x x < 1 Solution : Rearranging, we have : 2 2x x (1 + x) (1 x) 1 < x ( 1 x 2) (1 + x) (1 x) < 0 x2 + x + 4 (1 + x) (1 x) < 0

8 OpenStax-CNX module: m Now, polynomial in the numerator i.e. x 2 + x + 4 is positive for all real x as D<0 and a>0. Thus, dividing either side of the inequality by this polynomial does not change inequality. Now, we need to change the sign of x in one of the linear factors of the denominator positive in accordance with sign rule. This is required to be done in the factor (1-x). For this, we multiply each side of inequality by -1. This change in sign accompanies change in inequality as well : 1 (1 + x) (1 x) > 0 Critical points are -1 and 1. Hence, solution of the inequality in x is : Sign diagram Figure 8: Sign of function alternates. x (, 1) (1, ) 3.1 Rational inequality with repeated linear factors We have already discussed rational polynomial with repeated factors. We need to count repeated factors which appear in both numerator and denominator. If the linear factors are repeated even times, then we do not need to change sign about critical point corresponding to repeated linear factor. Note : While working with rational function having repeated factors, we need to factorize higher order polynomial like cubic polynomial. In such situation, we can employ a short cut. We guess one real root of the cubic polynomial. We may check corresponding equation with values such as 1,2, -1 or -2 etc and see whether cubic expression becomes zero or not for that value. If one of the roots is known, then cubic expression is f(x) = (x-a) g(x), where "a" is the guessed root and g(x) is a quadratic expression. We can then nd other two roots anlayzing quadratic expression. For example, x 3 6x x 6 = (x 1) ( x 2 5x + 6 ) = (x 1) (x 2) (x 3) Example 4 Problem : Find interval of x satisfying the inequality given by : (2x + 1) (x 1) (x 3 3x 2 + 2x) 0 Solution : We factorize each of the polynomials in numerator and denominator : (2x + 1) (x 1) (2x + 1) (x 1) (x 3 3x 2 = + 2x) x (x 1) (x 2)

9 OpenStax-CNX module: m It is important that we do not cancel common factors or terms. Here, critical points are - 1/2,1,0,1 and 2. The critical point "1" is repeated even times. Hence, we do not change sign about "1" while drawing sign scheme. Sign diagram Figure 9: Sign of function alternates. While writing interval, we drop equality sign for critical points, which corresponds to denominator. 1/2 x < 0 2 < x < [ 1/2, 0) (2, ) We do not include "-1" and "1" as they reduce denominator to zero. 4 Polynomial inequality We can treat polynomial inequality as rational inequality, because a polynomial function is a rational function with denominator as 1. Logically, sign method used for rational function should also hold for polynomial function. Let us consider a simple polynomial inequality, f (x) = 2x 2 + x 1 < 0. Here, function is product of two linear factors (2x-3)(x+2). Clearly, x=3/2 and x=-2 are the critical points. The sign scheme of the function is shown in the gure : Sign diagram Figure 10: Sign of function alternates. Solution of x satisfying inequality is : x ( 2, 3 ) 2 It is evident that this method is easier and mechanical in approach.

11 OpenStax-CNX module: m Example 5 Problem : Find domain of the function : f (x) = {1 (1 x 2 )} Solution : One radical (inner) is contained with another radical (outer). For the outer radical, 1 1 x x 2 1 The term on each side of inequality is a positive quantity. Squaring each side does not change inequality, 1 x 2 1 x 2 0 This quadratic inequality is true for all real x. Now, for inner radical 1 x 2 0 (1 + x) (1 x) 0 We multiply by -1 to change the sign of x in 1-x, Using sign rule : (x + 1) (x 1) 0 x [ 1, 1] Since, conditions corresponding to two radicals need to be fullled simultaneously, the domain of the given function is intersection of outer and inner radicals.

12 OpenStax-CNX module: m Domain of the function Figure 11: Intersection of two domains. Domain = [ 1, 1] Example 6 Problem : Find the domain of the function given by : f (x) = (x 14 x 11 + x 6 x 3 + x 2 + 1) Solution : Clearly, function is real for values of x for which expression within square root is a non negative number. We note that independent variable is raised to positive integers. The nature of each monomial depends on the value of x and nature of power. If x 1, then monomial evaluates to higher value for higher power. If x lies between 0 and 1, then monomial evaluates to lower value for higher power. Further, a negative x yields negative value when raised to odd power and positive value when raised to even power. We shall use these properties to evaluate the expression for three dierent intervals of x. x 14 x 11 + x 6 x 3 + x We consider dierent intervals of values of expression for dierent values of x, which cover the complete interval of real numbers. 1: x 1 In this case, x a > x b, if a > b. Evaluating in groups, ( x 14 x 11) + ( x 6 x 3) + ( x ) > 0 2: 0 x < 1 In this case, x a < x b, if a > b. Rearranging in groups, x 14 { ( x 11 x 6) + ( x 3 x 2) } + 1 Here, { ( x 11 x 6) + ( x 3 x 2) } is negative. Hence total expression is positive,

13 OpenStax-CNX module: m : x < 0 Rearranging in groups, x 14 { ( x 11 x 6) + ( x 3 x 2) } + 1 > 0 ( x 14 x 11) + ( x 6 x 3) + ( x ) Here, x 14, x 6 and x 2 are positive and x 11 and x 3 is negative. Hence, total expression is positive, ( x 14 x 11) + ( x 6 x 3) + ( x ) > 0 We see that expression is positive for all values of x. Hence, domain of the function is : Domain = R = (, ) 6 Exercise Exercise 1 (Solution on p. 14.) Find solution of the rational inequality given by : (x + 1) (x + 5) (x 3) Exercise 2 (Solution on p. 14.) Find solution of the rational inequality given by : 0 8x x 51 (2x 3) (x + 4) > 0 Exercise 3 (Solution on p. 14.) Find solution of the rational inequality given by : x 2 + 4x + 3 x 3 6x x 6 > 0 Exercise 4 (Solution on p. 14.) Find solution of the rational inequality given by : (2x + 1) (x 1) 2 (x 3 3x 2 + 2x) 0

14 OpenStax-CNX module: m Solutions to Exercises in this Module Solution to Exercise (p. 13) Hint : Critical points are -5,-1 and 3. We need to exclude end corresponding to x=3 as denominator turns zero for this value. Solution to Exercise (p. 13) Hint : Critical points are -4,-3,3/2,5/2. [ 5, 1] (3, ) x (, 4) ( 3, 3/2) (5/2, ) Solution to Exercise (p. 13) Hint : Factorize denominator as x 3 6x 2 +11x 6 = (x 1) (x 2) (x 3). Critical points are -3,-1,1,2,3. Solution to Exercise (p. 13) Hint : Factorize denominator as x ( 3, 1) (1, 2) (3, ) (2x + 1) (x 1)2 (x 3 3x 2 + 2x) = (2x + 1) (x 1)2 x (x 1) (x 2) Critical points are -1/2,1,1,0,1 and 2. We see that "1" is repeated odd times. Hence, we continue to assign alternating signs in accordance with wavy curve method. The solution of x for the inequality is : < x 1/2 0 < x < 1 2 < x <

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### Algebraic Expressions and Equations: Classification of Expressions and Equations

OpenStax-CNX module: m21848 1 Algebraic Expressions and Equations: Classification of Expressions and Equations Wade Ellis Denny Burzynski This work is produced by OpenStax-CNX and licensed under the Creative

### Florida Department of Education/Office of Assessment January 2012. Algebra 1 End-of-Course Assessment Achievement Level Descriptions

Florida Department of Education/Office of Assessment January 2012 Algebra 1 End-of-Course Assessment Achievement Level Descriptions Algebra 1 EOC Assessment Reporting Category Functions, Linear Equations,

### is the degree of the polynomial and is the leading coefficient.

Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...

### Indiana 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

### Zeros of Polynomial Functions

Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### 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

### 3.3 Real Zeros of Polynomials

3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section

### This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

### Contents. Introduction and Notes pages 2-3 (These are important and it s only 2 pages ~ please take the time to read them!)

Page Contents Introduction and Notes pages 2-3 (These are important and it s only 2 pages ~ please take the time to read them!) Systematic Search for a Change of Sign (Decimal Search) Method Explanation

### Solving Quadratic & Higher Degree Inequalities

Ch. 8 Solving Quadratic & Higher Degree Inequalities We solve quadratic and higher degree inequalities very much like we solve quadratic and higher degree equations. One method we often use to solve quadratic

### 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.

### 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

### 2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

### Vocabulary 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

### 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

### Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### OpenStax-CNX module: m32633 1. Quadratic Sequences 1; 2; 4; 7; 11;... (1)

OpenStax-CNX module: m32633 1 Quadratic Sequences Rory Adams Free High School Science Texts Project Sarah Blyth Heather Williams This work is produced by OpenStax-CNX and licensed under the Creative Commons

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

### 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

### Lecture 7 : Inequalities 2.5

3 Lecture 7 : Inequalities.5 Sometimes a problem may require us to find all numbers which satisfy an inequality. An inequality is written like an equation, except the equals sign is replaced by one of

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### Algebra 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 real-life problem. algebraic expression - An expression with variables.

### Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities

Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned

### 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

### The degree of a polynomial function is equal to the highest exponent found on the independent variables.

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### Developmental Math Course Outcomes and Objectives

Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/Pre-Algebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and

### SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

### 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

### 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

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.

### 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

### Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

### 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

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

### 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

### ModuMath Algebra Lessons

ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations

### Pre Cal 2 1 Lesson with notes 1st.notebook. January 22, Operations with Complex Numbers

0 2 Operations with Complex Numbers Objectives: To perform operations with pure imaginary numbers and complex numbers To use complex conjugates to write quotients of complex numbers in standard form Complex

### In this lesson you will learn to find zeros of polynomial functions that are not factorable.

2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has

### 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

### 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

### Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left.

Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote

### 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

### Mth 95 Module 2 Spring 2014

Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression

### Mathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework

Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal

### ALGEBRA I / ALGEBRA I SUPPORT

Suggested Sequence: CONCEPT MAP ALGEBRA I / ALGEBRA I SUPPORT August 2011 1. Foundations for Algebra 2. Solving Equations 3. Solving Inequalities 4. An Introduction to Functions 5. Linear Functions 6.

### 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

### Course Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)

Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,

### 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.

### 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.

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### 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

### 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

### 2.5 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 x-axis and

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

### Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.

Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson

### Copy 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,...}

### Review for Calculus Rational Functions, Logarithms & Exponentials

Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

### PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

### Algebra 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

1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal

### 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

### 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

### 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

### 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

### Algebra 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,

### MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

### 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,

### 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

### 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

### 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

### 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

### a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2

Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0

### 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

### POLYNOMIAL 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

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

### LINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL

Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables

### Algebra 2 Notes AII.7 Functions: Review, Domain/Range. Function: Domain: Range:

Name: Date: Block: Functions: Review What is a.? Relation: Function: Domain: Range: Draw a graph of a : a) relation that is a function b) relation that is NOT a function Function Notation f(x): Names the

### Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

### Systems of Linear Equations: Two Variables

OpenStax-CNX module: m49420 1 Systems of Linear Equations: Two Variables OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section,

### 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

### 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

### Solving Quadratic Equations by Completing the Square

9. Solving Quadratic Equations by Completing the Square 9. OBJECTIVES 1. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a geometric application

### Chapter 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

### Domain of a Composition

Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### 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

### Solutions to Self-Test for Chapter 4 c4sts - p1

Solutions to Self-Test for Chapter 4 c4sts - p1 1. Graph a polynomial function. Label all intercepts and describe the end behavior. a. P(x) = x 4 2x 3 15x 2. (1) Domain = R, of course (since this is a

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### 3.4 Complex Zeros and the Fundamental Theorem of Algebra

86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

### Logo Symmetry Learning Task. Unit 5

Logo Symmetry Learning Task Unit 5 Course Mathematics I: Algebra, Geometry, Statistics Overview The Logo Symmetry Learning Task explores graph symmetry and odd and even functions. Students are asked to

Polynomials and Quadratics Want to be an environmental scientist? Better be ready to get your hands dirty!.1 Controlling the Population Adding and Subtracting Polynomials............703.2 They re Multiplying

### Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative