Directions Please read carefully!


 Reynard Stafford
 1 years ago
 Views:
Transcription
1 Math Xa Algebra Practice Problems (Solutions) Fall 2008 Directions Please read carefully! You will not be allowed to use a calculator or any other aids on the Algebra PreTest or PostTest. Be sure to write neatly illegible answers will receive little or no credit. You will have one hour to complete the each test, which will consist of questions. Good luck! You can review for material tested below by looking at the corresponding review materials online. Solving linear equations 1. Solve P = a + art for t. t = P a ar 2. Solve A = 2lw + 2lh + 2w for h. A 2lw 2w = 2lh A 2lw 2w = h 2l A = 2lw + 2lh + 2w Solving quadratic equations 3. Solve = 0. Use quadratic formula. The solution is = 1 ± 15/3
2 4. Solve 2/3 1/3 6. This is a quadratic equation in disguise. Let u = 1/3. Then the equations becomes u 2 u 6 = 0 Factor to find the solutions u = 2 and u = 3. Then solve 1/3 = 2 by cubing both sides to get = 8 and solve 1/3 = 3 by cubing to get = 27. Solving polynomial equations by factoring 5. Solve p 3 = pmn 2 for p. p 3 = pmn 2 p 3 pmn 2 = 0 p(p 2 mn 2 ) = 0 From here we get the solution p = 0 and the equation p 2 mn 2 = 0 which we can solve either using the quadratic formula or rewriting the equation as p 2 = mn 2 so that p = ± mn. Thus there are three solutions p = 0 and p = ± mn. Solving equations involving rational epressions 6. Solve + 2(+3) 2 5 = 2 Page 2
3 (+3) (+3) 2 5 = 2 = ( + 3) = = 2 10 = 2(9 + 12) 8 = 24 = 3 7. Solve = The common denominator here is (4 2 + )( 1) = (4 + 1)( 1) = 0 ( 1) (4 2 + )( 1) + 2(4 2 + ) ( 1)(4 2 + ) 3( 1) (4 + 1)( 1) = 0 ( 1) + 2(4 2 + ) 3( 1) = 0 (4 + 1)( 1) ) = 0 (4 + 1)( 1) (4 + 1)( 1) = 0 2(3 + 2) (4 + 1)( 1) = 0 Remember that a rational epression p()/q() is equal to zero for that make p() = 0 and q() 0. The numerator of our fraction is equal to zero at = 0 and = 2/3. But = 0 also makes the denominator zero, so this cannot be a solution. Thus the only solutions is = 2/3. Page 3
4 Solving linear inequalities 8. Solve the inequality > 2 3. > Solve the inequality 3 10 < < 9 2 or ( 3 4, 9 2 ] Solving polynomial inequalities 10. Solve the inequality > 0. First we factor the lefthand side to get ( 4)( + 5) > 0. So the lefthand side is equal to zero when = 4 and = 5. We know that ( 4)( + 5) is positive if and only if either both factors are positive or both factors are negative. Both factors are positive if > 4 and > 5 these two inequalities are both true if > 4. Both factors are negative if < 4 and < 5 these two inequalities are both true if < 5. Thus we have two regions on which our inequality is satisfied: > 4 or < 5, which we can also write as (, 5) (4, ). We can do this same analysis by noting that = 4 and = 5 are the only places where = 0 and since f() = is a continuous function we know that for all values of in ( infty, 5), f() has the same sign. To find the sign for this region, we just test a point f( 6) = 10 > 0, hence the epression is positive on (, 5). For the region ( 5, 4) we can test = 0 giving f(0) = 20 < 0 so that the epression is negative on this region. And for the region (4, ) we test f(5) = 10 > 0 and so the epression is positive on this region. This gives the same solution as above. 11. Solve the inequality 4 < 2. Page 4
5 First we get everything over to one side (and zero on the other side) and then we factor. 4 2 < 0 2 ( 2 1) < 0 2 ( 1)( + 1) < 0 4 < 2 Thus 4 2 = 0 when = 0, 1, or 1. This splits the number line into four regions on which the sign of the epression 4 2 is constant: (, 1), ( 1, 0), (0, 1), (1, ). We test points on each of these regions to find that 4 2 < 0 on the regions ( 1, 0) and (0, 1). Note that since we want the epression to be strictly less than zero, we cannot combine these two regions together! The solution is ( 1, 0) (0, 1). Solving rational inequalities 12. Find the solution to the inequality > 0 The epression is zero when = 2 and undefined when = 1. This splits the number line into three regions on which the sign of 2 is constant: (, 1), ( 1, 2), (2, ). Testing +1 points in each of these regions, we find that 2 is positive in the regions (, 1) +1 and (2, ). The solution is (, 1) (2, ). Note that 2 is not a solution to the inequality because it makes the epression zero, whereas 1 is not a solution to the inequality because it makes the epression undefined. 13. Find the solution to the inequality First, we get everything over to the lefthand side, with zero on the right Page 5
6 hand side. Net, we get a common denominator, in this case ( 4) The numerator is zero when = 7 and the denominator is zero when = 4. This splits the number line into three regions on which the sign of 3+21 is constant: 4 (, 4), (4, 7), (7, ). Testing points in each of these regions we find that 3+21 is 4 negative on the intervals (, 4) and (7, ). Since = 7 makes the epression zero and = 4 makes the epression undefined, we have the solution (, 4) [7, ). Solving absolutevalue inequalities 14. Solve the inequality t Remember that for any mathematical epression A(t), A(t) > c means that either A(t) < c or A(t) > c (and similarly A(t) < c means that c < A(t) < c). Thus in this case we have two inequalities t or t The first inequality has solution t 3 5, the second has solution t. The full 2 2 solution set is t or t, that is t (, ] [ 3, ) Solve the inequality + r c, where c is a positive real number. Page 6
7 Remember that for any mathematical epression A(t), A(t) < c means that c < A(t) < c (and similarly A(t) > c means that either A(t) < c or A(t) > c). Thus in this case we have the combination inequality c + r c c r c r Simplifying epressions involving eponents and radicals 16. Simplify the following epression as completely as possible: ( c3 d 2 ) 5 3d ( c3 d 2 ) 5 3d = d 3 c 3/2 It would also be acceptable to give the answer ( c3 d 2 ) 5 3d = /2 d 3 c 3/2 17. Simplify the following epression as completely as possible: c 3 d 2 4cd 5 d 2 Any of the following three forms would be acceptable solutions c 3 d 2 4cd 5 d 2 = c 3 d 4 4cd 3 = c 3 d 4 4c d 3 = c3 d 7 4c d 3 Page 7
8 Simplifying polynomial epressions 18. Factor the following epression as completely as possible z 2 y 9 + 2z 3 y 8 + z 4 y 7 z 2 y 9 + 2z 3 y 8 + z 4 y 7 = z 2 y 7 (y 2 + 2zy + z 2 ) = z 2 y 7 (y + z) Epand and then simplify as much as possible 2 ( ) ( 21 11) ( ) ( 21 11) = = = 2 ( ) Either of the last two lines would be considered a complete simplification. Simplifying rational epressions 20. Simplify the following + y 2 + y 2 First, we reinterpret the negative eponents as fractions. Net we try to get rid of the fractions within fractions by clearing denominators. Then we Page 8
9 simplify if possible (in this case there is no further simplification possible). + y 2 + y 2 = + y y 2 = + y y 2 2 y2 2 y 2 = ( + y)(2 y 2 ) y Simplify the following assuming 0. ( ) ( ) ( 4 7 ) ( ) + 3 = 2 3 2( + 3) Simplify the following, assuming We can use a strategy of combining the numerator into one fraction and combining the denominator into one fraction, and then instead of division we invert and multiply = = = 1 2 Page 9
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
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More information2. Simplify. College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses
College Algebra Student SelfAssessment 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
More informationName Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE
Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers
More informationGuide 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 65 43 21 0
More informationIntegration of Rational Expressions by Partial Fractions
Integration of Rational Epressions by Partial Fractions INTRODUTION: We start with a few definitions rational epression is formed when a polynomial is divided by another polynomial In a proper rational
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationGRE MATH REVIEW #5. 1. Variable: A letter that represents an unknown number.
GRE MATH REVIEW #5 Eponents and Radicals Many numbers can be epressed as the product of a number multiplied by itself a number of times. For eample, 16 can be epressed as. Another way to write this is
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationMathematics Higher Tier, Algebraic Fractions
These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or an organisation and would like to purchase these solutions please contact Chatterton
More informationRational Expressions and Rational Equations
Rational Epressions and Rational Equations 6 6. Rational Epressions and Rational Functions 6. Multiplication and Division of Rational Epressions 6. Addition and Subtraction of Rational Epressions 6.4 Comple
More informationMath Common Core Sampler Test
High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests
More informationSimplification Problems to Prepare for Calculus
Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.
More informationSimplifying Exponential Expressions
Simplifying Eponential Epressions Eponential Notation Base Eponent Base raised to an eponent Eample: What is the base and eponent of the following epression? 7 is the base 7 is the eponent Goal To write
More information1.2 Linear Equations and Rational Equations
Linear Equations and Rational Equations Section Notes Page In this section, you will learn how to solve various linear and rational equations A linear equation will have an variable raised to a power of
More informationDefinition 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 informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More information3.1. 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
More informationA positive exponent means repeated multiplication. A negative exponent means the opposite of repeated multiplication, which is repeated
Eponents Dealing with positive and negative eponents and simplifying epressions dealing with them is simply a matter of remembering what the definition of an eponent is. division. A positive eponent means
More informationThe numerical values that you find are called the solutions of the equation.
Appendi F: Solving Equations The goal of solving equations When you are trying to solve an equation like: = 4, you are trying to determine all of the numerical values of that you could plug into that equation.
More informationChapter 3 Section 6 Lesson Polynomials
Chapter Section 6 Lesson Polynomials Introduction This lesson introduces polynomials and like terms. As we learned earlier, a monomial is a constant, a variable, or the product of constants and variables.
More informationIntegrating algebraic fractions
Integrating algebraic fractions Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate
More informationMethod 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
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationEquations 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.
More informationZero: 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 informationSIMPLIFYING ALGEBRAIC FRACTIONS
Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is
More information2.3. 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
More informationMathematics 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
More informationMATH 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
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More informationAlgebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials
More informationUnit 1: Polynomials. Expressions:  mathematical sentences with no equal sign. Example: 3x + 2
Pure Math 0 Notes Unit : Polynomials Unit : Polynomials : Reviewing Polynomials Epressions:  mathematical sentences with no equal sign. Eample: Equations:  mathematical sentences that are equated with
More informationSolving 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,
More informationStudents 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
More information5.4 The Quadratic Formula
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
More informationExponent Law Review 3 + 3 0. 12 13 b. 1 d. 0. x 5 d. x 11. a 5 b. b 8 a 8. b 2 a 2 d. 81u 8 v 10 81. u 8 v 20 81. Name: Class: Date:
Name: Class: Date: Eponent Law Review Multiple Choice Identify the choice that best completes the statement or answers the question The epression + 0 is equal to 0 Simplify 6 6 8 6 6 6 0 Simplify ( ) (
More informationIntegrals 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 informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationSubstitute 4 for x in the function, Simplify.
Page 1 of 19 Review of Eponential and Logarithmic Functions An eponential function is a function in the form of f ( ) = for a fied ase, where > 0 and 1. is called the ase of the eponential function. The
More informationSolving Logarithmic Equations
Solving Logarithmic Equations Deciding How to Solve Logarithmic Equation When asked to solve a logarithmic equation such as log (x + 7) = or log (7x + ) = log (x + 9), the first thing we need to decide
More information( yields. Combining the terms in the numerator you arrive at the answer:
Algebra Skillbuilder Solutions: 1. Starting with, you ll need to find a common denominator to add/subtract the fractions. If you choose the common denominator 15, you can multiply each fraction by one
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + 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
More informationSUNY 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)
More informationA Yearlong Pathway to Complete MATH 1111: College Algebra
A Yearlong Pathway to Complete MATH 1111: College Algebra A yearlong path to complete MATH 1111 will consist of 12 Learning Support (LS) classes and MATH 1111. The first semester will consist of the
More informationDevelopmental Math Course Outcomes and Objectives
Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/PreAlgebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationPolynomial and Rational Functions
Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest
More informationAN EASY LOOK AT THE CUBIC FORMULA
1 AN EASY LOOK AT THE CUBIC FORMULA Thomas J. Osler Mathematics Department Rowan University Glassboro NJ 0808 Osler@rowan.edu Introduction All students learn the quadratic formula for finding the roots
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions  that is, algebraic fractions  and equations which contain them. The reader is encouraged to
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationExamples of Tasks from CCSS Edition Course 3, Unit 5
Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can
More informationRational Expressions  Complex Fractions
7. Rational Epressions  Comple Fractions Objective: Simplify comple fractions by multiplying each term by the least common denominator. Comple fractions have fractions in either the numerator, or denominator,
More information27 = 3 Example: 1 = 1
Radicals: Definition: A number r is a square root of another number a if r = a. is a square root of 9 since = 9 is also a square root of 9, since ) = 9 Notice that each positive number a has two square
More informationFactoring 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 informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationAccuplacer Elementary Algebra Study Guide for Screen Readers
Accuplacer Elementary Algebra Study Guide for Screen Readers The following sample questions are similar to the format and content of questions on the Accuplacer Elementary Algebra test. Reviewing these
More informationHFCC Math Lab Intermediate Algebra  7 FINDING THE LOWEST COMMON DENOMINATOR (LCD)
HFCC Math Lab Intermediate Algebra  7 FINDING THE LOWEST COMMON DENOMINATOR (LCD) Adding or subtracting two rational expressions require the rational expressions to have the same denominator. Example
More informationChapter 7  Roots, Radicals, and Complex Numbers
Math 233  Spring 2009 Chapter 7  Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More information6.2 Properties of Logarithms
6. Properties of Logarithms 437 6. Properties of Logarithms In Section 6., we introduced the logarithmic functions as inverses of eponential functions and discussed a few of their functional properties
More informationa) 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
More informationLinear Equations and Inequalities
Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109  Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationMath 115 SelfAssessment Test is designed to assist you to determine how ready you are to take Math 115 (PreCalculus) at UNBC.
Math/Stats Math 115 SelfAssessment Test is designed to assist you to determine how ready you are to take Math 115 (PreCalculus) at UNBC. Contents 1. About Math 115. How to take the Self Assessment Test.
More information5.1 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
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE:  Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
More informationMath 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =
Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could
More information1 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 informationExponents, Radicals, and Scientific Notation
General Exponent Rules: Exponents, Radicals, and Scientific Notation x m x n = x m+n Example 1: x 5 x = x 5+ = x 7 (x m ) n = x mn Example : (x 5 ) = x 5 = x 10 (x m y n ) p = x mp y np Example : (x) =
More informationMULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.
1.4 Multiplication and (125) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with
More informationNegative Integer Exponents
7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More informationMultiplying and Dividing Algebraic Fractions
. Multiplying and Dividing Algebraic Fractions. OBJECTIVES. Write the product of two algebraic fractions in simplest form. Write the quotient of two algebraic fractions in simplest form. Simplify a comple
More informationACCUPLACER Arithmetic & Elementary Algebra Study Guide
ACCUPLACER Arithmetic & Elementary Algebra Study Guide Acknowledgments We would like to thank Aims Community College for allowing us to use their ACCUPLACER Study Guides as well as Aims Community College
More informationexpression 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 informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationLINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,
LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are
More information3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes
Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same
More informationDMA 060 Polynomials and Quadratic Applications
DMA 060 Polynomials and Quadratic Applications Brief Description This course provides a conceptual study of problems involving graphic and algebraic representations of quadratics. Topics include basic
More informationEquations Involving Fractions
. Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationSimplifying Algebraic Fractions
5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions
More information6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms
AAU  Business Mathematics I Lecture #6, March 16, 2009 6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms 6.1 Rational Inequalities: x + 1 x 3 > 1, x + 1 x 2 3x + 5
More informationBrunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year.
Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year. Goal The goal of the summer math program is to help students
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationSolving Rational Equations and Inequalities
85 Solving Rational Equations and Inequalities TEKS 2A.10.D Rational functions: determine the solutions of rational equations using graphs, tables, and algebraic methods. Objective Solve rational equations
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationTic Tac Toe Review or Partner Activity
Tic Tac Toe Math Factoring Practice & Solving Quadratics by Factoring Mied Problems HSA.REI.B.4b  Solve HSA.SSE.B.3a  Factor HSA.IF.C.8  Factor Tic Tac Toe Review or Partner Activity Grade Level: Algebra
More informationUnit 3 Polynomials Study Guide
Unit Polynomials Study Guide 75 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial
More informationMohawk Valley Community College MVCC MA115 Mr. Bauer
Mohawk Valley Community College MVCC MA115 Course description: This is a dual credit course. Successful completion of the course will give students 1 VVS Credit and 3 MVCC Credit. College credits do have
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More information