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

Size: px
Start display at page:

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

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 even-index radical) of a negative number. Can t take the logarithm of zero or a negative number. Two common notations to write the domain are set-builder and interval notation.. Set-builder 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.

Chapter 7 - Roots, Radicals, and Complex Numbers

Chapter 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 information

Exponents, Radicals, and Scientific Notation

Exponents, 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 information

1.3 Algebraic Expressions

1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

More information

1.6 The Order of Operations

1.6 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 information

Algebra 1 Course Title

Algebra 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 information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

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

More information

2.3. Finding polynomial functions. An Introduction:

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

More information

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Algebraic 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 information

Unit 7: Radical Functions & Rational Exponents

Unit 7: Radical Functions & Rational Exponents Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving

More information

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have 8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

More information

Vocabulary Words and Definitions for Algebra

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

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

Simplification of Radical Expressions

Simplification of Radical Expressions 8. Simplification of Radical Expressions 8. OBJECTIVES 1. Simplify a radical expression by using the product property. Simplify a radical expression by using the quotient property NOTE A precise set of

More information

5.1 Radical Notation and Rational Exponents

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

More information

Algebra and Geometry Review (61 topics, no due date)

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

More information

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) 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,

More information

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

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

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

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

More information

Higher Education Math Placement

Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

Lesson 9: Radicals and Conjugates

Lesson 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 information

Order of Operations More Essential Practice

Order of Operations More Essential Practice Order of Operations More Essential Practice We will be simplifying expressions using the order of operations in this section. Automatic Skill: Order of operations needs to become an automatic skill. Failure

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

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,...}

More information

COLLEGE ALGEBRA. Paul Dawkins

COLLEGE 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 information

0.8 Rational Expressions and Equations

0.8 Rational Expressions and Equations 96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

More information

3.1. RATIONAL EXPRESSIONS

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

More information

Alum 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 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 information

Radicals - Multiply and Divide Radicals

Radicals - Multiply and Divide Radicals 8. Radicals - Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals

More information

Review of Intermediate Algebra Content

Review 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 information

2.6 Exponents and Order of Operations

2.6 Exponents and Order of Operations 2.6 Exponents and Order of Operations We begin this section with exponents applied to negative numbers. The idea of applying an exponent to a negative number is identical to that of a positive number (repeated

More information

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

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)

More information

Lesson 9: Radicals and Conjugates

Lesson 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 information

Algebra Practice Problems for Precalculus and Calculus

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

More information

LAKE ELSINORE UNIFIED SCHOOL DISTRICT

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:

More information

HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING 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 information

Exponents and Radicals

Exponents 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 information

Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome 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 information

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.

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. 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 information

Fractions and Linear Equations

Fractions and Linear Equations Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

8-6 Radical Expressions and Rational Exponents. Warm Up Lesson Presentation Lesson Quiz

8-6 Radical Expressions and Rational Exponents. Warm Up Lesson Presentation Lesson Quiz 8-6 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 information

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

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

More information

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

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

More information

Florida 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 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 information

9.3 OPERATIONS WITH RADICALS

9.3 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 information

POLYNOMIAL FUNCTIONS

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

More information

Partial 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: 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 information

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 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 information

Simplifying Square-Root Radicals Containing Perfect Square Factors

Simplifying Square-Root Radicals Containing Perfect Square Factors DETAILED SOLUTIONS AND CONCEPTS - 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!

More information

Algebra I Vocabulary Cards

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

More information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

More information

Substitute 4 for x in the function, Simplify.

Substitute 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 information

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

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

More information

Answers to Basic Algebra Review

Answers 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 information

4.1. COMPLEX NUMBERS

4.1. 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 information

SIMPLIFYING SQUARE ROOTS

SIMPLIFYING SQUARE ROOTS 40 (8-8) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify

More information

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

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

More information

Radicals - Rational Exponents

Radicals - Rational Exponents 8. Radicals - Rational Exponents Objective: Convert between radical notation and exponential notation and simplify expressions with rational exponents using the properties of exponents. When we simplify

More information

Radicals - Rationalize Denominators

Radicals - Rationalize Denominators 8. Radicals - Rationalize Denominators Objective: Rationalize the denominators of radical expressions. It is considered bad practice to have a radical in the denominator of a fraction. When this happens

More information

ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES

ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES 1. Squaring a number means using that number as a factor two times. 8 8(8) 64 (-8) (-8)(-8) 64 Make sure students realize that x means (x ), not (-x).

More information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

Section 5.0A Factoring Part 1

Section 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 information

A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents

A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify

More information

MATH 10034 Fundamental Mathematics IV

MATH 10034 Fundamental Mathematics IV MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

More information

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

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

More information

Mathematics Placement

Mathematics Placement Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

More information

Multiplying and Dividing Radicals

Multiplying and Dividing Radicals 9.4 Multiplying and Dividing Radicals 9.4 OBJECTIVES 1. Multiply and divide expressions involving numeric radicals 2. Multiply and divide expressions involving algebraic radicals In Section 9.2 we stated

More information

Factoring Polynomials and Solving Quadratic Equations

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

More information

Determinants can be used to solve a linear system of equations using Cramer s Rule.

Determinants 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 information

A Quick Algebra Review

A Quick Algebra Review 1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

More information

Big Bend Community College. Beginning Algebra MPC 095. Lab Notebook

Big 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 information

Solutions of Linear Equations in One Variable

Solutions 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 information

Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:

Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions: Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?

More information

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students

More information

What are the place values to the left of the decimal point and their associated powers of ten?

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

More information

A Systematic Approach to Factoring

A 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 information

PREPARATION FOR MATH TESTING at CityLab Academy

PREPARATION FOR MATH TESTING at CityLab Academy PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST

More information

Zeros of Polynomial Functions

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

More information

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

More information

MBA Jump Start Program

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

More information

Radicals - Multiply and Divide Radicals

Radicals - Multiply and Divide Radicals 8. Radicals - Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals

More information

Core Maths C1. Revision Notes

Core 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 information

Answer Key for California State Standards: Algebra I

Answer Key for California State Standards: Algebra I Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

More information

Solving Rational Equations and Inequalities

Solving Rational Equations and Inequalities 8-5 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 information

Negative Integer Exponents

Negative 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 information

Equations, Inequalities & Partial Fractions

Equations, Inequalities & Partial Fractions Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

More information

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 9 Order of Operations

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 9 Order of Operations Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 9 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm

More information

Indices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková

Indices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková Indices and Surds The term indices refers to the power to which a number is raised. Thus x is a number with an index of. People prefer the phrase "x to the power of ". Term surds is not often used, instead

More information

0.4 FACTORING POLYNOMIALS

0.4 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 information

Five 5. Rational Expressions and Equations C H A P T E R

Five 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 information

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

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

More information

Figure 1. A typical Laboratory Thermometer graduated in C.

Figure 1. A typical Laboratory Thermometer graduated in C. SIGNIFICANT FIGURES, EXPONENTS, AND SCIENTIFIC NOTATION 2004, 1990 by David A. Katz. All rights reserved. Permission for classroom use as long as the original copyright is included. 1. SIGNIFICANT FIGURES

More information

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

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

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

Algebra Cheat Sheets

Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

More information

SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property

SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property 498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

NSM100 Introduction to Algebra Chapter 5 Notes Factoring Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

More information

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

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

More information

MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

MULTIPLICATION 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 (1-25) 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 information

Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as:

Limits. 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 information

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

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

More information