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


 Delphia Miller
 2 years ago
 Views:
Transcription
1 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 denominators. Find domain of a radical function. Evaluate expressions involving rational exponents. Solve a radical equation. Evaluate square roots or cube roots. Example: Evaluate 9 This is a square root. The number under the radical is 9. (An asterisk * means multiply.) 9 Find a number such that the number squared is 9. This number is because 9 Example: Evaluate 8 This is a cube root. The number under the radical is 8. 8 Find a number such that the number cubed is 8. This number is because 8.
2 Page of Example: Evaluate 400 This is a square root. The number under the radical is does not have a real number solution. Find a number such that the number squared is 400. Note that and (not 400). Any real number squared is a positive number or zero. Example: Evaluate 400 This is a square root. The number under the radical is Find a number such that the number squared is 400. ( ) Thus 0 0 Don t forget the negative sign in front of. Example: Evaluate 7 This is a cube root. The number under the radical is 7. ( ) 7 Find a number such that the number cubed will give you 7. The number is because ( ) ( ) ( ) ( ) 7.
3 Page of Simplify radical expressions and Rationalize denominators Usually to simplify means to rewrite the expression in such a way that it has as few radicals as possible, and that the expression under each radical does not contain perfect powers. Some rules for radicals are illustrated below. a. or ( ) b. *0 0 c d. 5 * 7 5*7 5 e *5 0 Also, ( 7 ) 7 7 and 5 (+ 5) 6 * Keep in mind that: does not equal 5 + 7, and 5 5 does not equal. 7 7 Example: Simplify 8. The only thing which can be simplified is 8 under the square root. Can it be factored in such a way that one of the multiples is a perfect square? 8 4*7 *7 * 7 7 Factor 8 so one factor is a perfect square. The radical of a product equals the product of the radicals. Simplify the perfect square.
4 Page 4 of Example: Simplify 7x. Assume x 0. The expression under the square root has a perfect square. 7x 7 x 7 x The radical of a product equals the product of the radicals. Simplify the perfect square. Example: Simplify 5 k. Assume k 0. The radical is a square root. The expression under the square root is not a perfect square. 5 4 k k k 4 k k k k. Factor so there are perfect squares. The radical of a product equals the product of the radicals. Simplify the perfect square. k remains under the radical since the exponent on k is which is smaller than the index () of the radical Example: Simplify 7 8 k. Assume k 0 and q 0. 0 q The radical is a square root. The expression under the square root is not a perfect square kq 0 k q 5 64 k k q k k q 5 8 k k q kq k Factor so there are perfect squares. The radical of a product equals the product of the radicals. Simplify the perfect squares. The radical is simplified because there are no perfect squares left under the radical and the remaining variables and numbers have exponents of one.
5 Page 5 of Example: Simplify 7k 5. Assume k 0. This is a cube root. Each factor is a perfect cube. and k ( k ) 7k 5 5 The radical of a product equals the product of k 7 5 k 5 ( k ) the radicals. Write each factor as a perfect cube. Simplify the cube roots. Example: Simplify x y. Assume x 0 and y 0. The radical is a cube root. The expression under the cube root is not a perfect cube. 64x y x y The radical of a product is the product of the 4 4 x xy xy x x x x y x y y y y y radicals. Factor so there are perfect cubes. Simplify the perfect cubes. Rearrange the factors. Generally the radical factor is written last The radical is simplified because there are no perfect cubes left in the radical.
6 Page 6 of Example: Simplify t 8t. Assume t 0. The expression is the product of two square roots. t 8t t 8t The product of two radicals equals the radical of their product. 4t Multiply. 4 t The radical of the product equals the product of the radicals. 6 t Factor so there are perfect squares. Simplify the square roots and rearrange. t 6 Example: Simplify y 4 y + 5. Assume y. The expression is the product of two square roots. y 4 y 4 The product of two radicals equals the radical of the product. y + 5 y + 5 ( y ) 4 ( y + 5) ( y ) y + 5 y y + 5 Multiply. Factor the perfect square in the numerator. Distribute in the denominator. Simplify the perfect square.
7 Page 7 of Example: Simplify 6. 6 The expression is the quotient of two cube roots The quotient of two radicals equals the radical of their quotient. Divide. Simplify the perfect cube. 7 Example: Simplify The expression is the sum of two radicals. The radicals are like radicals. Both contain (4 + 8) 5 Add the like radicals. 5 Example: Simplify The expression contains the sum of four radical terms. The like radicals can be added Rearrange so like radicals are together. ( 4 + 9) 5 + ( + ) Combine the like radicals
8 Page 8 of Example: Simplify The expression contains the sum of two square roots Simplify the radicals to determine if there are like radicals Combine the like radicals. 7 Example: Simplify x 8 + x 8. Assume x 8. The expression contains the sum of two terms containing radicals. x 8 + x 8 4 x 8 The radicals in both terms are like radicals ( x 8 ), so the terms can be added. Example: Simplify x + + 9x + 8. Assume x. The expression contains the sum of two terms. x + + 9x + 8 x + + 9( x + ) Simplify the radicals. x + + x + + x + 4 x + ( x + ) Combine the like radicals.
9 Page 9 of Example: Rationalize the denominator. 5 Rationalize the denominator means to find an equivalent fraction whose denominator does not contain a radical ( 5) Multiply the numerator and the denominator by 5. (Note: You can t just square the numerator and the denominator. It will change the value of the fraction) Simplify the denominator.
10 Page 0 of Example: Rationalize denominator The denominator of this expression is irrational because it includes an irrational number ( ). Rationalize denominator means to find an equivalent expression, but with a rational denominator. 5 (5 )(5 ) 5+ (5+ )(5 ) ( ) There is a binomial in the denominator. Multiply the numerator and denominator by ( 5 ). (The conjugate of the denominator.) Find the product of the numerator and denominator. Combine like radicals. Note: Choosing ( 5 ) uses the formula ( a b)( a b) a b and b. Squaring The fraction would change the value of the fraction. +. In this example, a 5
11 Page of Find the domain of a function Domain The domain is a list or set of all possible inputs that yield a real number output. There are three operations we can t do with real numbers in algebra. Each of these restrict the domain. Can t divide by zero. Can t take the square root (or any evenindex radical) of a negative number. Can t take the logarithm of zero or a negative number. Two common notations to write the domain are setbuilder and interval notation.. Setbuilder notation: Sets are typically written in braces { }. The notation is { independent variable some property or restriction about independent variable } where the vertical line is read such that. Example: All real numbers, x, less than. { xx< } Example: All real numbers, n 4 n< 6 { } n, greater than or equal to 4 and less than 6.. Interval notation: Parenthesis indicate the starting or ending value is not included and a square bracket indicates the starting or ending value is included. Within the parentheses or square bracket, we indicate the smallest value of x followed by a comma and then the largest value of x. The examples above are shown using interval notation. Example: All real numbers, x, less than., ( ) Example: All real numbers, 4,6 [ ) n, greater than or equal to 4 and less than 6.
12 Page of Find the domain of a radical function Example: Find the domain of y x. The function contains a square root. The expression under the square root, x, must be greater or equal to zero. x 0 x x 6 In interval notation the answer is [ 6, ) Isolate the term with a variable. Divide both sides by. ( is positive, so don t change the inequality sign) Example: Find the domain of y 5 t. The function contains a square root. The expression under the square root, 5 t, must be greater or equal to zero. 5 t 0 t 5 5 t In interval notation the answer is 5,. Isolate the term with a variable. Divide both sides by. (Remember when you multiply or divide an inequality by a negative number, the inequality sign changes direction.)
13 Page of Example: Find the domain of y x + 4. The function contains cube root. The expression under the cube root can be any real number. x + 4 x + 4 can be any real number ( ) (, ) Write the expression under the radical. (The expression is called the radicand.) Since the radical is a cube root the expression can be any real number. Write the domain using interval notation. Evaluate expressions involving rational exponents For the problems in this group, an expression containing rational exponents should be written using radical notation, and an expression containing radical notation should be written using rational exponents. n n The definition of rational exponent is x x n The definition of a negative exponent is x. n x All rules for exponents apply to rational and negative exponents. Often used rules are listed below. Assume: a 0 m n m n Product Rule: a a a + m Power Rule: ( a ) n m a a mn m n Quotient Rule: a n a Example: Write 7 using rational exponents. The expression contains a cube root, which could be rewritten using rational exponents. 7 7 ( ) 7 7 * Rewrite the radical using a rational exponent. Use the power rule.
14 Page 4 of Example: Evaluate 8. The exponent is, which is a rational number. Use the definition for rational exponents. 8 8 (8 ) ( 8) () 4 Use the power rule. The definition of a rational exponent is used. Simplify the cube root. Note: If you use a calculator, remember to use parentheses. Enter 8^(/), not 8^/.
15 Page 5 of Example: Evaluate 9 The exponent is, which is a negative rational number ( 9 ) ( 9 ) ( ) 7 Use the definition of a negative exponent. Use the power rule. Write the power as square root. Simplify the square root and raise the result to the third power. If you use a calculator, remember to use parentheses. Enter 9^(/), not 9^/
16 Page 6 of Solve radical equations A radical equation is an equation containing one or more radical terms. For example, x x is a radical equation. To solve means to determine all the real values which, when substituted in the equation for x, will make the statement true. All such real values should be included in the answer. Note that 0 0 is true. Hence, x 0 is a solution of the equation x x. It turns out that x is also a solution of x x verify that This second solution is often ignored. To find all the solutions, follow the steps for solving radical equations given below. For equations containing one radical, the steps are:. Isolate the radical.. Square both sides of the equation. Solve the resulting equation which no longer contains radicals. This equation is often linear, quadratic, or rational. 4. Check the answers. (It is possible that some x values may be in the solution set of the resulting equation, but will not make the original equation true.) For equations with two radicals:. Isolate one of the radicals.. Square both sides of the equation.. Combine like terms. 4. Now the equation either has no radicals, or just one radical term. a. If there are no radicals follow steps and 4 under equations containing one radical. b. If there is one radical follow steps through 4 under equations containing one radical.
17 Page 7 of Example: Solve the equation x + 6 This is an equation (contains an equal sign). Use the steps for equations containing one radical. x + 6 ( x + ) 6 x + 6 x + 6 x 4 The radical is isolated. Square both sides. Solve the resulting linear equation. Check Substitute x 4 into the original equation and simplify the results. The statement is true so the solution is x 4. Example: Solve the equation x + 6 This is an equation (contains an equal sign). Use the steps for equations containing one radical. x + 6 ( x + ) ( 6) x + 6 x 4 The radical is isolated. Square both sides. Solve the resulting linear equation. Check Substitute x 4 into the original equation and simplify the results. The statement is false, so x 4 is not a solution. Note: The square root of any real number can t be negative. Hence, x + 6 can t be true.
18 Page 8 of Example: Solve the equation x + 4 x 40. This is an equation (contains an equal sign). It contains two radical terms. x + 4 x 40 5 x 40 x 8 ( ) x 8 x x.5 Add like radicals. Follow the steps for equations containing one radical. Divide both sides by 5. The radical is isolated. Square both sides. Solve the resulting linear equation. Check ( ) ( ) ( ) Substitute x into the original equation and simplify the results. 65 The statement is true so x is a solution to the equation.
19 Page 9 of Example: Solve the equation x x 0 This is an equation (contains an equal sign). It contains one radical term. Check x x 0 x x ( x ) ( x) x 4x x x 4 0 x( 4 x) 0 x 0 or ( 4x ) 0 x 0 or 4x x 0 or x 4 Check 0 Check x : ( ) x : Isolate the radical. Square both sides; remember to square each factor. This is a quadratic equation that can be solved by factoring. Set each factor equal to zero. Solve the two linear equations. There are two possible answers. Substitute each answer into the original equation. Simplify the results to be sure each yields a true statement. Both answers check.
20 Page 0 of Example: Solve the equation x + x This is an equation (contains an equal sign). It contains one radical term. x + x x + ( x ) x + x 6x + 9 x 7x x x 6 0 ( )( ) x 0 or x 6 0 x or x 6 Check Check x : + 4 Check x 6 : The radical is isolated. Square both sides. Note: ( x ) x. This is a quadratic equation that can be solved by factoring. Set each factor equal to zero. Solve the two linear equations. There are two possible solutions. Substitute each answer into the original equation. Simplify the results to be sure each yields a true statement. Notice x does not yield a true statement, while x 6 yields a true statement. The solution is x 6.
21 Page of Example: Solve the equation x + x 4 This is an equation (contains an equal sign). It contains two radical terms. Use the steps for equations that contain two radicals. x+ x 4 x ( x )( x ) ( x ) ( x ) ( ) 4 x x x x+ + x 4 x+ + 4 x x 4+ x 4 x 4 6 x 4 ( x ) 6 x 4 ( x ) x 4 x ( 4) x ( x 5) 0 or ( x 8) 0 x 5 or x 8 Isolate one radical by adding x 4 to both sides. Square both sides. + x 4 + x 4 Note: ( ) Now there is one radical term left in the equation. Combine like terms and isolate the radical. Factor two out of the terms on the right side. Divide both sides of the equation by a common factor of. Squared both sides. This is a quadratic equation that can be solved by factoring. Set each factor equal to zero. Solve the two linear equations. There are two possible solutions.
22 Page of Check Check 5 Check 8 x : ( ) x : ( ) Substitute each answer into the original equation. Simplify the results to be sure each yields a true statement. Both x 5 and x 8 check, so they are both solutions.
1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More 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 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 informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More 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 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 informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More 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 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 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 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 informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More information86 Radical Expressions and Rational Exponents. Warm Up Lesson Presentation Lesson Quiz
86 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt Algebra ALgebra2 2 Warm Up Simplify each expression. 1. 7 3 7 2 16,807 2. 11 8 11 6 121 3. (3 2 ) 3 729 4. 5.
More informationWelcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move
More informationEAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors.
EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section
ALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 53.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 64.2 Solving Equations by
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationCollege Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More information4.1. COMPLEX NUMBERS
4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More 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 informationDeterminants can be used to solve a linear system of equations using Cramer s Rule.
2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution
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 informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
More informationA Systematic Approach to Factoring
A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool
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 informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More 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 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 informationTool 1. Greatest Common Factor (GCF)
Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
More informationMBA 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
More informationSECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31
More information0.4 FACTORING POLYNOMIALS
36_.qxd /3/5 :9 AM Page 9 SECTION. Factoring Polynomials 9. FACTORING POLYNOMIALS Use special products and factorization techniques to factor polynomials. Find the domains of radical expressions. Use
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationMath 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction
Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a
More informationDomain 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
More informationSouth Carolina College and CareerReady (SCCCR) Algebra 1
South Carolina College and CareerReady (SCCCR) Algebra 1 South Carolina College and CareerReady Mathematical Process Standards The South Carolina College and CareerReady (SCCCR) Mathematical Process
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationCAHSEE on Target UC Davis, School and University Partnerships
UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,
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 informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More 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 informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationAlgebra 2 YearataGlance Leander ISD 200708. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 YearataGlance Leander ISD 200708 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More 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 informationUnit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials
Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial
More informationLimits. Graphical Limits Let be a function defined on the interval [6,11] whose graph is given as:
Limits Limits: Graphical Solutions Graphical Limits Let be a function defined on the interval [6,11] whose graph is given as: The limits are defined as the value that the function approaches as it goes
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationNSM100 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
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationReview of Basic Algebraic Concepts
Section. Sets of Numbers and Interval Notation Review of Basic Algebraic Concepts. Sets of Numbers and Interval Notation. Operations on Real Numbers. Simplifying Expressions. Linear Equations in One Variable.
More informationSOLVING QUADRATIC EQUATIONS  COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen
SOLVING QUADRATIC EQUATIONS  COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen A. GENERALITIES. When a given quadratic equation can be factored, there are 2 best methods
More informationPOLYNOMIALS and FACTORING
POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use
More information1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style
Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationis the degree of the polynomial and is the leading coefficient.
Property: T. HrubikVulanovic email: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 HigherDegree Polynomial Functions... 1 Section 6.1 HigherDegree Polynomial Functions...
More informationBookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina  Beaufort Lisa S. Yocco, Georgia Southern University
More informationFACTORING POLYNOMIALS
296 (540) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated
More informationA. Factoring out the Greatest Common Factor.
DETAILED SOLUTIONS AND CONCEPTS  FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
More informationFACTORING QUADRATIC EQUATIONS
FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods
More informationPROBLEMS AND SOLUTIONS  OPERATIONS ON IRRATIONAL NUMBERS
PROBLEMS AND SOLUTIONS  OPERATIONS ON IRRATIONAL NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More informationAlgebra I Credit Recovery
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
More informationSolving 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
More informationFinding Solutions of Polynomial Equations
DETAILED SOLUTIONS AND CONCEPTS  POLYNOMIAL EQUATIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More informationFactoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1)
Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if
More informationAlgebra 2: Q1 & Q2 Review
Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More 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 informationAlgebra 1 Advanced Mrs. Crocker. Final Exam Review Spring 2014
Name: Mod: Algebra 1 Advanced Mrs. Crocker Final Exam Review Spring 2014 The exam will cover Chapters 6 10 You must bring a pencil, calculator, eraser, and exam review flip book to your exam. You may bring
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More informationFINAL 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
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationAlgebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test
Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationPrerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.
Course Syllabus Math 1314 College Algebra Revision Date: 82115 Catalog Description: Indepth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems
More informationLyman Memorial High School. PreCalculus Prerequisite Packet. Name:
Lyman Memorial High School PreCalculus Prerequisite Packet Name: Dear PreCalculus Students, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. These
More informationPolynomial Operations and Factoring
Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationAIP Factoring Practice/Help
The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More informationSuccessful completion of Math 7 or Algebra Readiness along with teacher recommendation.
MODESTO CITY SCHOOLS COURSE OUTLINE COURSE TITLE:... Basic Algebra COURSE NUMBER:... RECOMMENDED GRADE LEVEL:... 811 ABILITY LEVEL:... Basic DURATION:... 1 year CREDIT:... 5.0 per semester MEETS GRADUATION
More informationMTH124: Honors Algebra I
MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,
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 informationThis is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).
This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationAlgebra II New Summit School High School Diploma Program
Syllabus Course Description: Algebra II is a two semester course. Students completing this course will earn 1.0 unit upon completion. Required Materials: 1. Student Text Glencoe Algebra 2: Integration,
More informationList the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated
MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible
More informationBEGINNING ALGEBRA ACKNOWLEDMENTS
BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science
More information6.1 The Greatest Common Factor; Factoring by Grouping
386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.
More informationSUFFOLK COMMUNITY COLLEGE MATHEMATICS AND COMPUTER SCIENCE DEPARTMENT STUDENT COURSE OUTLINE Fall 2011
SUFFOLK COMMUNITY COLLEGE MATHEMATICS AND COMPUTER SCIENCE DEPARTMENT STUDENT COURSE OUTLINE Fall 2011 INSTRUCTOR: Professor Emeritus Donald R. Coscia OFFICE: Online COURSE: MAT111Algebra II SECTION &
More information