POLYNOMIALS and RATIONAL EXPRESSIONS
|
|
|
- Mervin Gallagher
- 3 years ago
- Views:
Transcription
1 POLYNOMIALS and RATIONAL EXPRESSIONS Simplifying Rational Expressions Overview of Objectives, students should be able to: 1. Factor using sum and difference of cubes pattern.. Simply rational expressions via factoring. A C 3. Multiply and simplify rational expressions: B D 4. Divide and simplify two rational expressions: A C B D 5. Add and subtract two rational expressions with common denominators. Objectives: Main Overarching Questions: 1. How do you know you need to simplify an expression?. What do you do to simplify expressions? 3. For the sum and difference of cubes, how do you remember the formulas? Note: The rational operation and simplification objectives are similar to those in MAT 004, however problems are usually more involved and rely more on factoring. If students are familiar with the basic operations a general discussion about how the process works might be more appropriate than reintroducing the material from scratch. Activities and Questions to ask students: Factor using sum and difference of cubes pattern. Provide students with an example of the difference of SQUARES first: How did we factor 4x 9? How did we factor 16x 4 49y? Recall when we multiplied: (3x+y)(3x-y) We recognized the pattern as (a + b)(a - b) = a - b How could you use this information to factor 4x 9? (x-3)(x+3) Is it factored completely? Now, let s look at the sum and difference of CUBES: o a 3 b 3 = (a - b)(a + ab + b ) o a 3 + b 3 = (a + b)(a - ab + b ) What differences do you notice between the two formulas? How can you prove the formulas are true? Talk through until they agree that multiplying the right side out will give them the sum/difference of cubes. How would you recognize the sum or difference of cubes? Ask students to provide an example of what the sum of cubes would look like. Ask how do they know it is the sum of cubes. Ask students to provide an example of what the difference of cubes would look like.
2 Give students a worksheet to work in groups. This worksheet should include all the possible factoring problems. Discuss answers. Have students give answers and defend their answers. Always ask is it factored completely? Simply rational expressions via factoring. Note: This is a review from MAT 004. However, the students should be better skilled at factoring. 16 How do you reduce the fraction? What process or operations did you use in your reduction? 30 Multiply and simplify rational expressions: What about 3 x x x x = 4 x x x x x x =? What process or operations did you use in your reduction? How do you know when it is reduced to lowest terms? Give another example: how could you simplify x 6x + 4x + 1x example. What processes do you need in order to simplify a rational expression? How could you simplify x x x + 5x + 4 What happens to the numerator when simplifying:?? Remember how you simplified the last 5 x x 5 8? What about x 7x + 10 x Do you see a pattern in the last two examples? (i.e. when the numerator disappears we must write a 1). Give a few simplification problems requiring factoring differences of squares, product/sum, and quadratic trinomials with leading coefficients not equal to 1. If students have trouble simplifying, review the process above. A C 4 7 How do you multiply fractions? i.e. B D 5 8 Give students several numeric examples. Do you see a pattern or process to multiply fractions? P R P R Have students draw out the process that =. Then, simplify. Q S Q S x 5 Have students work the example: x / Was there a shortcut to our first example? = = = / 5 10?
3 How can we use the shortcut to multiply: Have students work the example: 6y y + 1 x 6 4? 3 x 6 1 y 3y + y Divide and simplify two rational expressions: 3 7 How do you divide fractions?? i.e. A C 4 8 B D Give students several numeric examples. Do you see a pattern or process to divide fractions? P R P S P S Have students draw out the process that = =. Then, simplify. Q S Q R Q R Add and subtract two rational expressions with common denominators. Give students an example to use this process like: y y + 4y 1 y + 3y 8 y + 14y y 3 If you want to use the shortcut from multiplication, at what step would you perform it? 3 1 How do you add fractions with the same denominator? i.e Give students several numeric examples. Do you see a pattern or process to add fractions? 5 1 How do you subtract fractions with the same denominator? i.e. 8 8 Give students several numeric examples. Do you see a pattern or process to subtract fractions? P R P ± R Have students draw out the process that ± =. Then, simplify. Q Q Q 3y 6 Have students work out the example: y + 3y 10 y + 3y 10 Have students work out the example: x + 6x + x 1 x + x 6 x + x 6
4 Adding and Subtracting Rational Expressions with Unlike Denominators Overview of Objectives, students should be able to: 1. Find the least common denominator given two or more rational expressions. Add and subtract two or more rational expressions with different denominators Main Overarching Questions: 1. How do you find the LCD of a list of rational expressions?. How do you add or subtract rational expressions that contain different denominators? 3. Why do you have to get common denominators before adding/subtracting rational expressions? 4. How is adding/subtracting rational expressions similar to adding/subtracting fractions? Objectives: Find the least common denominator given two or more rational expressions Add and subtract two or more rational expressions with different denominators Activities and Questions to ask students: 1 How do you find the least common denominator?,. 3 6 Give several numeric examples. Do you see a pattern or process? 3 What is the least common denominator of,? Write down the process you used to find x y the least common denominator. 4x What about finding the LCD of,. If students try to piece together terms to get x 5 x 5 the LCD, remind them to think about the process they used to find the LCD with numbers (i.e. factoring) Give several more examples involving binomial and trinomial denominators. When is this process useful? For what purpose could we use this? 3? How do you write equivalent fractions? Give a numeric example: =. 4 8 Give several numeric examples. Do you see a pattern or process? Ask students what a cross product is. Can we use this to solve the problem another way? Do we get the same answer?
5 5? Have students work the example: 3y = 1y Does using cross products work in this case? When can we use cross products? When could you use this process to add or subtract fractions that do not have the same denominator? 3x? Have students work another example, rewrite = x + 5 x 5 1 How do you add fractions with different denominators? Give several numeric examples. Do you see a pattern or process? How can you use what you have already learned to add or subtract fractions with different denominators? 3 4 Have students work the example: +. Write down the process you used to solve the x 4x problem. 4 3 y + 9 Give a harder example to work: +, followed by: x x 1 y 7 y + 1 y 3 Have students work in groups on a worksheet of problems containing a mixture of addition, subtract, multiplication, and division problems. Is there more than one way to solve each problem? Explain. Complex Fractions and Polynomial Division Overview of Objectives, students should be able to: 1. Simplify complex rational expressions. Divide two polynomials a. Divide two polynomials by simplifying the resulting rational expression b. Divide two polynomials by using long division Objectives: Main Overarching Questions: 1. How do you simplify complex fractions?. How do you divide polynomials? Activities and Questions to ask students:
6 Simplify complex rational expressions Divide two polynomials o Divide two polynomials by simplifying the resulting rational expression o Divide two polynomials by using long division x Present students with a simpler complex fraction:. Ask students to write down what 4x 5 fractions they see in the complex fraction. Ask students what operation is represented between the two fractions (i.e. the fraction bar represents division) P R P S P S Then ask students to think about we divided fractions. = = Q S Q R Q R Have students summarize the process x y What if we have a more complicated complex fraction like? 1 1 x y Ask students to describe the differences between this complex fraction and the first one. How many fractions do I have in the numerator? How many in the denominator? How many did I have before? Once students realize there is now more than one fraction in the numerator and denominator, ask them how they might combine the two fractions in the numerator into one fraction. Ask students to recall what concepts they have studied that allows them to combine fractions with addition and subtraction. Ask students to summarize the process (i. e. combine all fractions in the numerator/denominator using LCD, write as division of two fractions, convert to multiplication and simplify). Give a group worksheet of one or two simpler complex fractions and one or two of the more complicated fractions. 4 5x Ask students to add the two fractions together: +. Hopefully they will notice the 5x 5x 4+ 5x common denominators and simply add the numerators:. 5x
7 Next ask them how they might simplify x. Is there a way to break up the fractions? If 8x students have trouble, ask them to consider how we combined the two fractions together from before. How do we reverse the process? Have students draw the conclusion that A + B = A + B C C C Give students a few practice problems to try. Ask students to next consider what happens if we have more than one term in the x 7x+ 1 denominator like. Can we split up the denominator? If students say yes, ask x 5 them to combine together 3 x +, using a common denominator. Ask them once again to 5 consider if they can split up the denominator. Once students are convinced that they cannot split the denominator, suggest using long division to divide. Give students all the terminology before beginning. As a good analogy, complete the steps to long division concurrently with an example from arithmetic. Have students compare/contrast each step of polynomial long division with the arithmetic version. Give students a few long division problems to try on their own. Solve Rational Equations Overview of Objectives, students should be able to: 1. Solve Rational Equations Objectives: Solve rational equations. Main Overarching Questions: 1. How do you solve rational equations?. What is an extraneous solution? Activities and Questions to ask students: 1 3 Give students a simple example of a rational equation: + =. x x Ask students to observe what value of x is not allowed. If they are not sure, says we are never allowed to have blank in the denominator.
8 Next ask them how they might proceed to solve the equation. What is different about this equation when compared with other equations we have solved? (We have fractions) How can we clear our fractions? What one quantity can we multiply by to ensure all fractions are cleared? Ask students what the LCD is in the previous example 1 + = 3. x x Do we have to multiply the by the LCD? Why? Have students multiply through by LCD to clear the fraction. What kind of equation is left over? How do we solve it? Have students summarize the process of solving radical equations. x 3 Give students another rational equation: 9 x3 = x3 +. Ask them to solve it. What solution did you get? Does this solution work when plugged in? How could we have known the solution would not work? Define the solution x = 3 in the previous example as an extraneous solution. Remind students to always list out restrictions on the variable at the beginning of the problem.
Polynomial and Rational Functions
Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving
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,
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
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
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
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
Section 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
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
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.
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
expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
north seattle community college
INTRODUCTION TO FRACTIONS If we divide a whole number into equal parts we get a fraction: For example, this circle is divided into quarters. Three quarters, or, of the circle is shaded. DEFINITIONS: The
POLYNOMIALS 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
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
HFCC Math Lab Arithmetic - 4. Addition, Subtraction, Multiplication and Division of Mixed Numbers
HFCC Math Lab Arithmetic - Addition, Subtraction, Multiplication and Division of Mixed Numbers Part I: Addition and Subtraction of Mixed Numbers There are two ways of adding and subtracting mixed numbers.
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.
Solving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
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.
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
1.3 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.
2.3 Solving Equations Containing Fractions and Decimals
2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions
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,...}
CAHSEE 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,
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
( ) 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
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
Quadratics - Build Quadratics From Roots
9.5 Quadratics - Build Quadratics From Roots Objective: Find a quadratic equation that has given roots using reverse factoring and reverse completing the square. Up to this point we have found the solutions
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
Chapter R.4 Factoring Polynomials
Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x
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
Pre-Calculus II Factoring and Operations on Polynomials
Factoring... 1 Polynomials...1 Addition of Polynomials... 1 Subtraction of Polynomials...1 Multiplication of Polynomials... Multiplying a monomial by a monomial... Multiplying a monomial by a polynomial...
Integrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
6.4 Special Factoring Rules
6.4 Special Factoring Rules OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor a sum of cubes. By reversing the rules for multiplication
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
Multiplying and Dividing Algebraic Fractions
. Multiplying and Dividing Algebraic Fractions. OBJECTIVES. Write the product of two algebraic fractions in simplest form. Write the quotient of two algebraic fractions in simplest form. Simplify a comple
Tool 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
Adding and Subtracting Fractions. 1. The denominator of a fraction names the fraction. It tells you how many equal parts something is divided into.
Tallahassee Community College Adding and Subtracting Fractions Important Ideas:. The denominator of a fraction names the fraction. It tells you how many equal parts something is divided into.. The numerator
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
Sample Fraction Addition and Subtraction Concepts Activities 1 3
Sample Fraction Addition and Subtraction Concepts Activities 1 3 College- and Career-Ready Standard Addressed: Build fractions from unit fractions by applying and extending previous understandings of operations
MATH 108 REVIEW TOPIC 10 Quadratic Equations. B. Solving Quadratics by Completing the Square
Math 108 T10-Review Topic 10 Page 1 MATH 108 REVIEW TOPIC 10 Quadratic Equations I. Finding Roots of a Quadratic Equation A. Factoring B. Quadratic Formula C. Taking Roots II. III. Guidelines for Finding
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
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
Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test
Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action
is identically equal to x 2 +3x +2
Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3
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
Section 4.1 Rules of Exponents
Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells
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
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
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
Polynomial Expression
DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
In algebra, factor by rewriting a polynomial as a product of lower-degree polynomials
Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting
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
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
The Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes
Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same
Factoring (pp. 1 of 4)
Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common
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
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
Activity 1: Using base ten blocks to model operations on decimals
Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division
Chapter 3 Section 6 Lesson Polynomials
Chapter Section 6 Lesson Polynomials Introduction This lesson introduces polynomials and like terms. As we learned earlier, a monomial is a constant, a variable, or the product of constants and variables.
A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
Greatest Common Factor (GCF) Factoring
Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication
Math Common Core Sampler Test
High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests
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
AIP 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
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
FRACTIONS OPERATIONS
FRACTIONS OPERATIONS Summary 1. Elements of a fraction... 1. Equivalent fractions... 1. Simplification of a fraction... 4. Rules for adding and subtracting fractions... 5. Multiplication rule for two fractions...
Vieta s Formulas and the Identity Theorem
Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion
Factoring Quadratic Trinomials
Factoring Quadratic Trinomials Student Probe Factor x x 3 10. Answer: x 5 x Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials. Part 1 of the lesson consists
SIMPLIFYING ALGEBRAIC FRACTIONS
Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is
TRIGONOMETRY Compound & Double angle formulae
TRIGONOMETRY Compound & Double angle formulae In order to master this section you must first learn the formulae, even though they will be given to you on the matric formula sheet. We call these formulae
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!
Chapter 5. Rational Expressions
5.. Simplify Rational Expressions KYOTE Standards: CR ; CA 7 Chapter 5. Rational Expressions Definition. A rational expression is the quotient P Q of two polynomials P and Q in one or more variables, where
Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai
Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for
How To Factor Quadratic Trinomials
Factoring Quadratic Trinomials Student Probe Factor Answer: Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials Part 1 of the lesson consists of circle puzzles
Factoring Polynomials
Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring
Operations with Algebraic Expressions: Multiplication of Polynomials
Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the
Partial Fractions Decomposition
Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational
10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006
MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions Created January 7, 2006 Math 092, Elementary Algebra, covers the mathematical content listed below. In order
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
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) (
Simplification Problems to Prepare for Calculus
Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.
Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving
Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words
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
3.3 Addition and Subtraction of Rational Numbers
3.3 Addition and Subtraction of Rational Numbers In this section we consider addition and subtraction of both fractions and decimals. We start with addition and subtraction of fractions with the same denominator.
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) =
This is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
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:
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
5.4 Solving Percent Problems Using the Percent Equation
5. Solving Percent Problems Using the Percent Equation In this section we will develop and use a more algebraic equation approach to solving percent equations. Recall the percent proportion from the last
UNIT 5 VOCABULARY: POLYNOMIALS
2º ESO Bilingüe Page 1 UNIT 5 VOCABULARY: POLYNOMIALS 1.1. Algebraic Language Algebra is a part of mathematics in which symbols, usually letters of the alphabet, represent numbers. Letters are used to
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.
Using Proportions to Solve Percent Problems I
RP7-1 Using Proportions to Solve Percent Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving
Fractions to decimals
Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of
Factoring. Factoring Polynomial Equations. Special Factoring Patterns. Factoring. Special Factoring Patterns. Special Factoring Patterns
Factoring Factoring Polynomial Equations Ms. Laster Earlier, you learned to factor several types of quadratic expressions: General trinomial - 2x 2-5x-12 = (2x + 3)(x - 4) Perfect Square Trinomial - x
A Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.