The greatest common factor, or GCF, is the largest factor that two or more terms share.
|
|
- Maurice Jennings
- 5 years ago
- Views:
Transcription
1 Unit, Lesson Factoring Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product You can factor certain expressions by writing them as the product of factors The Zero Product Property states that if the product of two factors is 0, then at least one of the factors is 0 After setting a quadratic equation equal to 0, you can sometimes factor the quadratic expression and solve the equation by setting each factor equal to 0 The greatest common factor, or GCF, is the largest factor that two or more terms share You should always check to see if the terms of an expression have a greatest common factor before attempting to factor further The value of a for a quadratic expression in the form ax + bx + c is called the leading coefficient, or lead coefficient, because it is the coefficient of the term with the highest power To factor a trinomial with a leading coefficient of 1 in the form x + bx + c, find two numbers d and e that have a product of c and a sum of b The factored form of the expression will be (x + d)(x + e) When finding d and e, be careful with the signs The table that follows shows that the signs of d and e will be based on the signs of b and c Signs of b, c, d, and e b c d e Opposite signs; the number with the larger absolute value is positive Opposite signs; the number with the larger absolute value is negative You may be able to factor expressions with lead coefficients other than 1 in your head or by using guess-and-check Expressions with leading coefficients other than 1 in the form factored by grouping ax b c can sometimes be If you struggle to factor expressions in your head or by using guess-and-check, factoring by grouping is a more structured alternative Factor by Grouping 1 Begin by finding two numbers d and e whose product is ac and whose sum is b Rewrite the expression by replacing bx with dx + ex: ax d e c Factor the greatest common factor from ax dx 4 Factor the greatest common factor from e c 5 Factor the greatest common factor from the resulting expression MUnitLesson 1 6/8/018
2 Unit, Lesson Factoring (continued) A quadratic expression in the form ax ) b ( is called a difference of squares The difference of squares ax ) b ( can be written in factored form as ( a b)(a b) Some expressions cannot be factored These expressions are said to be prime Although the difference of squares is factorable, the sum of squares is prime For example, ( ) ( 5)( 5), but ( ) is not factorable Common Errors/Misconceptions forgetting to consider the signs of a, b, and c treating a leading coefficient other than 1 as if it were a 1 multiplying the terms of an expression given in factored form to solve an equation solving an expression that is not part of an equation confusing the difference of squares with the sum of squares Example 1 Factor x The leading coefficient is 1, so begin by finding two numbers whose product is ac 15 and whose sum is b 8 The numbers are and 5 because ( )( 5) 15 and ( 5) 8 Find the factors Use and 5 to find the factors The factors will be ( ) and ( 5) Write the expression as the product of its factors The factors are ( ) and ( 5), so the product of the factors is ( )( 5) The product should be the original expression ( )( 5) 815 Example Solve by factoring 1 Rewrite the equation so that all terms are on one side of the equation Original equation 9x 8 56 Add x to both sides 9x 64 0 Subtract 56 from both sides MUnitLesson 6/8/018
3 Unit, Lesson Factoring (continued) Factor the difference of squares The expression on the left side can be rewritten in the form ( ) 8 You can use this form to rewrite the expression as the difference of squares to factor the expression ( 8)( 8) 0 Use the Zero Product Property to solve Example The expression will equal 0 only when one of the factors is equal to 0 Set each factor equal to 0 and solve 8 0 or So, when Solve x 8 0 by factoring 8 or 8 1 Rewrite the equation so that all terms are on one side of the equation x x 8 0 Original equation Subtract 0 from both sides Find the factors The leading coefficient of the expression is 1, so begin by finding two numbers whose product is ac 0 and whose sum is b 8 The numbers are and 10 because ( )(10 ) 0 and 10 8 Therefore, the factors are ( ) and ( 10 ) Write the expression as the product of its factors ( )( 10 ) 0 4 Use the Zero Product Property to solve the equation The expression will equal 0 only when one of the factors is equal to 0 Set each factor equal to 0 and solve 0 or 10 0 or 10 So, x 8 0 when or 10 MUnitLesson 6/8/018
4 Unit, Lesson Factoring (continued) Example 4 Factor 10 by grouping 1 Find two numbers whose product is ac and whose sum is b The expression is in the form ax b c a, b 1, and c 10 ac ()( 10 ) 0 You need to find two numbers whose product is 0 and whose sum is 1 The numbers are 6 and 5 because ( 6)(5) 0 and Replace bx in the original equation The numbers found above have a sum of b, so you can replace bx with 10 Original equation ( 6x 5x ) 10 Rewritten equation 6x 5x Factor the greatest common factor of the two left-hand terms Do the same with the two right-hand terms The greatest common factor of the left-hand terms is ( ) 510 The greatest common factor of the right-hand terms is 5 ( ) 5( ) 4 Factor the greatest common factor of the expression Example 5 The greatest common factor of the expression is ( ) 10 ( )( 5) You have just factored by grouping You can check your work by multiplying the factors The product should be the original expression Solve 7x Factor out the greatest common factor of the expression The greatest common factor is 7 7x ( 910 ) 0 Find two numbers whose product is 10 and whose sum is 9 The numbers are 1 and 10 because ( 1)(10 ) 10 and Use these numbers to find the factors The factors are ( 1) and ( 10 ) 4 Write the expression as the product of its factors 7( 1)( 10 ) 0 MUnitLesson 4 6/8/018
5 Unit, Lesson Factoring (continued) Example 5 (continued) 5 Use the Zero Product Property to solve The expression will equal 0 only when one of the factors is equal to 0 The factor of 7 will never equal 0 Set each of the remaining factors equal to 0 and solve 1 0 or So, 7x when 1 or when 10 Example 6 Solve 8x Rewrite the equation so that all the terms are on one side of the equation 8x 18 5 Original equation 8x Subtract 5 from both sides Factor the equation The leading coefficient is not 1, so factor by grouping Begin by finding two numbers whose product is ac and whose sum is b a 8, b 18, and c 5 ac (8)( 5) 40 Find two numbers whose product is 40 and whose sum is 18 The numbers are 0 and because ( 0 )( ) 40 and 0 18 Replace bx in the original expression The numbers you found have a sum of b, so you can replace bx with 8x ( 0 x ) 5 0 (8 0x ) ( 5) 0 0x x Factor out the greatest common factor of the left-hand terms Do the same with the righthand terms The greatest common factor of the left-hand terms is 4x 4x( 5) ( 5) 0 The greatest common factor of the right-hand terms is 1 4x( 5) 1( 5) 0 Factor out the greatest common factor of the expression, ( 5) ( 5)( 41) 0 MUnitLesson 5 6/8/018
6 Unit, Lesson Factoring (continued) Example 6 (continued) Set each factor equal to 0 and solve x 5 0 or 4x 1 0 x 5 4x So, 8x 18 5 when 5 or when 1 4 MUnitLesson 6 6/8/018
A quadratic expression is a mathematical expression that can be written in the form 2
118 CHAPTER Algebra.6 FACTORING AND THE QUADRATIC EQUATION Textbook Reference Section 5. CLAST OBJECTIVES Factor a quadratic expression Find the roots of a quadratic equation A quadratic expression is
More informationSolving Quadratic Equations
Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic
More informationKEY CONCEPTS. Factoring is the opposite of expanding.
KEY CONCEPTS Factoring is the opposite of expanding. To factor simple trinomials in the form x 2 + bx + c, find two numbers such that When you multiply them, their product (P) is equal to c When you add
More informationMultiplication of Polynomials
Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is
More informationNever leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!
1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a
More informationMathB65 Ch 4 VII, VIII, IX.notebook. November 06, 2017
Chapter 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest
More informationAlgebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials
Algebra 1: Hutschenreuter Chapter 10 Notes Name 10.1 Adding and Subtracting Polynomials Polynomial- an expression where terms are being either added and/or subtracted together Ex: 6x 4 + 3x 3 + 5x 2 +
More informationSecondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics
Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction
Prerequisite Skills This lesson requires the use of the following skills: simplifying radicals working with complex numbers Introduction You can determine how far a ladder will extend from the base of
More informationTopic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3
Topic 7: Polynomials Table of Contents 1. Introduction to Polynomials. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Special Products of Binomials 5. Factoring Polynomials 6. Factoring
More informationCan there be more than one correct factorization of a polynomial? There can be depending on the sign: -2x 3 + 4x 2 6x can factor to either
MTH95 Day 9 Sections 5.5 & 5.6 Section 5.5: Greatest Common Factor and Factoring by Grouping Review: The difference between factors and terms Identify and factor out the Greatest Common Factor (GCF) Factoring
More informationLesson 3.5 Exercises, pages
Lesson 3.5 Exercises, pages 232 238 A 4. Calculate the value of the discriminant for each quadratic equation. a) 5x 2-9x + 4 = 0 b) 3x 2 + 7x - 2 = 0 In b 2 4ac, substitute: In b 2 4ac, substitute: a 5,
More informationUnit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions
CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.
More informationAdding and Subtracting Polynomials
Adding and Subtracting Polynomials Polynomial A monomial or sum of monomials. Binomials and Trinomial are also polynomials. Binomials are sum of two monomials Trinomials are sum of three monomials Degree
More informationIntroduction. Adding and Subtracting Polynomials
Introduction Polynomials can be added and subtracted like real numbers. Adding and subtracting polynomials is a way to simplify expressions. It can also allow us to find a shorter way to represent a sum
More information9-8 Completing the Square
In the previous lesson, you solved quadratic equations by isolating x 2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square. When
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 1: Interpreting Structure in Expressions Instruction
Prerequisite Skills This lesson requires the use of the following skills: evaluating expressions using the order of operations evaluating expressions for a given value identifying parts of an expression
More information7.2 Solving Quadratic Equations by Factoring
7.2 Solving Quadratic Equations by Factoring 1 Factoring Review There are four main types of factoring: 1) Removing the Greatest Common Factor 2) Difference of square a 2 b 2 3) Trinomials in the form
More informationAlgebra I Unit Report Summary
Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit - ( Ascend Default unit) 1. A01_01_01 H-A-B.1 Word Phrases As Algebraic Expressions 2. A01_01_02
More informationLesson 12: Overcoming Obstacles in Factoring
Lesson 1: Overcoming Obstacles in Factoring Student Outcomes Students factor certain forms of polynomial expressions by using the structure of the polynomials. Lesson Notes Students have factored polynomial
More informationMathB65 Ch 4 IV, V, VI.notebook. October 31, 2017
Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest
More informationUNIT 2 FACTORING. M2 Ch 11 all
UNIT 2 FACTORING M2 Ch 11 all 2.1 Polynomials Objective I will be able to put polynomials in standard form and identify their degree and type. I will be able to add and subtract polynomials. Vocabulary
More informationB.3 Solving Equations Algebraically and Graphically
B.3 Solving Equations Algebraically and Graphically 1 Equations and Solutions of Equations An equation in x is a statement that two algebraic expressions are equal. To solve an equation in x means to find
More informationSolving Equations Quick Reference
Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number
More informationUNIT 3 REASONING WITH EQUATIONS Lesson 2: Solving Systems of Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: graphing equations of lines using properties of equality to solve equations Introduction Two equations that are solved together
More informationSolving Linear Equations
Solving Linear Equations Golden Rule of Algebra: Do unto one side of the equal sign as you will do to the other Whatever you do on one side of the equal sign, you MUST do the same exact thing on the other
More informationFoundations of Math II Unit 5: Solving Equations
Foundations of Math II Unit 5: Solving Equations Academics High School Mathematics 5.1 Warm Up Solving Linear Equations Using Graphing, Tables, and Algebraic Properties On the graph below, graph the following
More information5.3. Polynomials and Polynomial Functions
5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a
More informationChapter One: Pre-Geometry
Chapter One: Pre-Geometry Index: A: Solving Equations B: Factoring (GCF/DOTS) C: Factoring (Case Two leading into Case One) D: Factoring (Case One) E: Solving Quadratics F: Parallel and Perpendicular Lines
More informationLESSON 7.2 FACTORING POLYNOMIALS II
LESSON 7.2 FACTORING POLYNOMIALS II LESSON 7.2 FACTORING POLYNOMIALS II 305 OVERVIEW Here s what you ll learn in this lesson: Trinomials I a. Factoring trinomials of the form x 2 + bx + c; x 2 + bxy +
More informationFind two positive factors of 24 whose sum is 10. Make an organized list.
9.5 Study Guide For use with pages 582 589 GOAL Factor trinomials of the form x 2 1 bx 1 c. EXAMPLE 1 Factor when b and c are positive Factor x 2 1 10x 1 24. Find two positive factors of 24 whose sum is
More informationFactoring Polynomials. Review and extend factoring skills. LEARN ABOUT the Math. Mai claims that, for any natural number n, the function
Factoring Polynomials GOAL Review and extend factoring skills. LEARN ABOUT the Math Mai claims that, for any natural number n, the function f (n) 5 n 3 1 3n 2 1 2n 1 6 always generates values that are
More informationReview Notes - Solving Quadratic Equations
Review Notes - Solving Quadratic Equations What does solve mean? Methods for Solving Quadratic Equations: Solving by using Square Roots Solving by Factoring using the Zero Product Property Solving by Quadratic
More informationMath 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2
Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is
More informationAssignment #1 MAT121 Summer 2015 NAME:
Assignment #1 MAT11 Summer 015 NAME: Directions: Do ALL of your work on THIS handout in the space provided! Circle your final answer! On problems that your teacher would show work on be sure that you also
More informationThe number part of a term with a variable part. Terms that have the same variable parts. Constant terms are also like terms.
Algebra Notes Section 9.1: Add and Subtract Polynomials Objective(s): To be able to add and subtract polynomials. Recall: Coefficient (p. 97): Term of a polynomial (p. 97): Like Terms (p. 97): The number
More informationUnit 3A: Factoring & Solving Quadratic Equations After completion of this unit, you will be able to
Unit 3A: Factoring & Solving Quadratic Equations After completion of this unit, you will be able to Learning Target #1: Factoring Factor the GCF out of a polynomial Factor a polynomial when a = 1 Factor
More informationAdding and Subtracting Polynomials
Adding and Subtracting Polynomials When you add polynomials, simply combine all like terms. When subtracting polynomials, do not forget to use parentheses when needed! Recall the distributive property:
More informationFactoring Trinomials of the Form ax 2 + bx + c, a 1
Factoring Trinomials of the Form ax 2 + bx + c, a 1 When trinomials factor, the resulting terms are binomials. To help establish a procedure for solving these types of equations look at the following patterns.
More informationMath 1302 Notes 2. How many solutions? What type of solution in the real number system? What kind of equation is it?
Math 1302 Notes 2 We know that x 2 + 4 = 0 has How many solutions? What type of solution in the real number system? What kind of equation is it? What happens if we enlarge our current system? Remember
More informationHONORS GEOMETRY Summer Skills Set
HONORS GEOMETRY Summer Skills Set Algebra Concepts Adding and Subtracting Rational Numbers To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference
More informationEquations in Quadratic Form
Equations in Quadratic Form MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: make substitutions that allow equations to be written
More informationSection 9.1: Add and Subtract Polynomials. The number part of a term with a variable part.
Algebra Notes Section 9.1: Add and Subtract Polynomials Objective(s): Recall: Coefficient (p. 97): Term of a polynomial (p. 97): Like Terms (p. 97): The number part of a term with a variable part. The
More informationSection 7.1 Quadratic Equations
Section 7.1 Quadratic Equations INTRODUCTION In Chapter 2 you learned about solving linear equations. In each of those, the highest power of any variable was 1. We will now take a look at solving quadratic
More informationSection 2.4: Add and Subtract Rational Expressions
CHAPTER Section.: Add and Subtract Rational Expressions Section.: Add and Subtract Rational Expressions Objective: Add and subtract rational expressions with like and different denominators. You will recall
More informationOrder of Operations Practice: 1) =
Order of Operations Practice: 1) 24-12 3 + 6 = a) 6 b) 42 c) -6 d) 192 2) 36 + 3 3 (1/9) - 8 (12) = a) 130 b) 171 c) 183 d) 4,764 1 3) Evaluate: 12 2-4 2 ( - ½ ) + 2 (-3) 2 = 4) Evaluate 3y 2 + 8x =, when
More informationSection 6.5 A General Factoring Strategy
Difference of Two Squares: a 2 b 2 = (a + b)(a b) NOTE: Sum of Two Squares, a 2 b 2, is not factorable Sum and Differences of Two Cubes: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 b 3 = (a b)(a 2 + ab + b
More informationYou try: What is the equation of the line on the graph below? What is the equation of the line on the graph below?
1 What is the equation of the line on the graph below? 2 3 1a What is the equation of the line on the graph below? y-intercept Solution: To write an equation in slope-intercept form, identify the slope
More information{ independent variable some property or restriction about independent variable } where the vertical line is read such that.
Page 1 of 5 Introduction to Review Materials One key to Algebra success is identifying the type of work necessary to answer a specific question. First you need to identify whether you are dealing with
More informationGeometry Summer Review Packet Page 1
June 017 Dear Geometry Students and Parents: Welcome to Geometry! For the 017-018 school year, we would like to focus your attention to the fact that many concepts from Algebra I are infused into Geometry.
More informationReview for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.
LESSON 6- Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in
More informationDON ROBERT B. ESTRELLA SR. NATIONAL HIGH SCHOOL Nagsaag, San Manuel, Pangasinan. (Effective Alternative Secondary Education) MATHEMATICS II
DON ROBERT B. ESTRELLA SR. NATIONAL HIGH SCHOOL Nagsaag, San Manuel, Pangasinan. (Effective Alternative Secondary Education) MATHEMATICS II Y X MODULE 1 Quadratic Equations BUREAU OF SECONDARY EDUCATION
More informationMULTIPLYING TRINOMIALS
Name: Date: 1 Math 2 Variable Manipulation Part 4 Polynomials B MULTIPLYING TRINOMIALS Multiplying trinomials is the same process as multiplying binomials except for there are more terms to multiply than
More informationUnit 5 AB Quadratic Expressions and Equations 1/9/2017 2/8/2017
Unit 5 AB Quadratic Expressions and Equations 1/9/2017 2/8/2017 Name: By the end of this unit, you will be able to Add, subtract, and multiply polynomials Solve equations involving the products of monomials
More informationBasic Equation Solving Strategies
Basic Equation Solving Strategies Case 1: The variable appears only once in the equation. (Use work backwards method.) 1 1. Simplify both sides of the equation if possible.. Apply the order of operations
More informationLesson 21 Not So Dramatic Quadratics
STUDENT MANUAL ALGEBRA II / LESSON 21 Lesson 21 Not So Dramatic Quadratics Quadratic equations are probably one of the most popular types of equations that you ll see in algebra. A quadratic equation has
More informationReview of Rational Expressions and Equations
Page 1 of 14 Review of Rational Epressions and Equations A rational epression is an epression containing fractions where the numerator and/or denominator may contain algebraic terms 1 Simplify 6 14 Identification/Analysis
More informationUnit 2 Quadratics. Mrs. Valentine Math 3
Unit 2 Quadratics Mrs. Valentine Math 3 2.1 Factoring and the Quadratic Formula Factoring ax 2 + bx + c when a = ±1 Reverse FOIL method Find factors of c that add up to b. Using the factors, write the
More informationPolynomials; Add/Subtract
Chapter 7 Polynomials Polynomials; Add/Subtract Polynomials sounds tough enough. But, if you look at it close enough you ll notice that students have worked with polynomial expressions such as 6x 2 + 5x
More informationUnit 7: Factoring Quadratic Polynomials
Unit 7: Factoring Quadratic Polynomials A polynomial is represented by: where the coefficients are real numbers and the exponents are nonnegative integers. Side Note: Examples of real numbers: Examples
More informationPrime Factorization and GCF. In my own words
Warm- up Problem What is a prime number? A PRIME number is an INTEGER greater than 1 with EXACTLY 2 positive factors, 1 and the number ITSELF. Examples of prime numbers: 2, 3, 5, 7 What is a composite
More informationChapter Six. Polynomials. Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring
Chapter Six Polynomials Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring Properties of Exponents The properties below form the basis
More information5-9. Complex Numbers. Key Concept. Square Root of a Negative Real Number. Key Concept. Complex Numbers VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING
TEKS FOCUS 5-9 Complex Numbers VOCABULARY TEKS (7)(A) Add, subtract, and multiply complex TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. Additional TEKS (1)(D),
More informationA-2. Polynomials and Factoring. Section A-2 1
A- Polynomials and Factoring Section A- 1 What you ll learn about Adding, Subtracting, and Multiplying Polynomials Special Products Factoring Polynomials Using Special Products Factoring Trinomials Factoring
More informationFactor each expression. Remember, always find the GCF first. Then if applicable use the x-box method and also look for difference of squares.
NOTES 11: RATIONAL EXPRESSIONS AND EQUATIONS Name: Date: Period: Mrs. Nguyen s Initial: LESSON 11.1 SIMPLIFYING RATIONAL EXPRESSIONS Lesson Preview Review Factoring Skills and Simplifying Fractions Factor
More informationPartial Fraction Decomposition Honors Precalculus Mr. Velazquez Rm. 254
Partial Fraction Decomposition Honors Precalculus Mr. Velazquez Rm. 254 Adding and Subtracting Rational Expressions Recall that we can use multiplication and common denominators to write a sum or difference
More informationAdditional Practice Lessons 2.02 and 2.03
Additional Practice Lessons 2.02 and 2.03 1. There are two numbers n that satisfy the following equations. Find both numbers. a. n(n 1) 306 b. n(n 1) 462 c. (n 1)(n) 182 2. The following function is defined
More informationAlgebra I Polynomials
Slide 1 / 217 Slide 2 / 217 Algebra I Polynomials 2014-04-24 www.njctl.org Slide 3 / 217 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying
More informationP.5 Solving Equations
PRC Ch P_5.notebook P.5 Solving Equations What you should learn How to solve linear equations How to solve quadratic equations equations How to solve polynomial equations of degree three or higher How
More informationMath 0320 Final Exam Review
Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:
More informationLesson 6. Diana Pell. Monday, March 17. Section 4.1: Solve Linear Inequalities Using Properties of Inequality
Lesson 6 Diana Pell Monday, March 17 Section 4.1: Solve Linear Inequalities Using Properties of Inequality Example 1. Solve each inequality. Graph the solution set and write it using interval notation.
More informationLesson 6: Switching Between Forms of Quadratic Equations Unit 5 Quadratic Functions
(A) Lesson Context BIG PICTURE of this UNIT: CONTEXT of this LESSON: How do we analyze and then work with a data set that shows both increase and decrease What is a parabola and what key features do they
More informationAlgebra I. Book 2. Powered by...
Algebra I Book 2 Powered by... ALGEBRA I Units 4-7 by The Algebra I Development Team ALGEBRA I UNIT 4 POWERS AND POLYNOMIALS......... 1 4.0 Review................ 2 4.1 Properties of Exponents..........
More informationBeginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions
1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:
More informationChapter 3: Section 3.1: Factors & Multiples of Whole Numbers
Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers Prime Factor: a prime number that is a factor of a number. The first 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
More informationSolving Quadratic Equations by Formula
Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always
More informationQUADRATIC FUNCTIONS AND MODELS
QUADRATIC FUNCTIONS AND MODELS What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results to sketch graphs of functions. Find minimum and
More informationAlgebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella
1 Algebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella In this summer assignment, you will be reviewing important topics from Algebra I that are crucial
More informationALGEBRA CLAST MATHEMATICS COMPETENCIES
2 ALGEBRA CLAST MATHEMATICS COMPETENCIES IC1a: IClb: IC2: IC3: IC4a: IC4b: IC: IC6: IC7: IC8: IC9: IIC1: IIC2: IIC3: IIC4: IIIC2: IVC1: IVC2: Add and subtract real numbers Multiply and divide real numbers
More informationQuadratic Formula: - another method for solving quadratic equations (ax 2 + bx + c = 0)
In the previous lesson we showed how to solve quadratic equations that were not factorable and were not perfect squares by making perfect square trinomials using a process called completing the square.
More informationAre you ready for Algebra 3? Summer Packet *Required for all Algebra 3/Trigonometry Students*
Name: Date: Period: Are you ready for Algebra? Summer Packet *Required for all Students* The course prepares students for Pre Calculus and college math courses. In order to accomplish this, the course
More informationEX: Simplify the expression. EX: Simplify the expression. EX: Simplify the expression
SIMPLIFYING RADICALS EX: Simplify the expression 84x 4 y 3 1.) Start by creating a factor tree for the constant. In this case 84. Keep factoring until all of your nodes are prime. Two factor trees are
More informationFactoring Review Types of Factoring: 1. GCF: a. b.
Factoring Review Types of Factoring: 1. GCF: a. b. Ex. A. 4 + 2 8 B. 100 + 25 2. DOS: a. b. c. Ex. A. 9 B. 2 32 3. Plain x Trinomials: Start Signs Factors 1. 2. 3. 4. Ex. A. + 7 + 12 B. 2 3 4. Non-Plain
More informationAlgebra I. Polynomials.
1 Algebra I Polynomials 2015 11 02 www.njctl.org 2 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying
More informationChapter 9 Quadratic Functions and Equations
Chapter 9 Quadratic Functions and Equations 1 9 1Quadratic Graphs and their properties U shaped graph such as the one at the right is called a parabola. A parabola can open upward or downward. A parabola
More informationPARTIAL FRACTION DECOMPOSITION. Mr. Velazquez Honors Precalculus
PARTIAL FRACTION DECOMPOSITION Mr. Velazquez Honors Precalculus ADDING AND SUBTRACTING RATIONAL EXPRESSIONS Recall that we can use multiplication and common denominators to write a sum or difference of
More informationNAME DATE PERIOD. Study Guide and Intervention. Solving Polynomial Equations. For any number of terms, check for: greatest common factor
5-5 Factor Polynomials Study Guide and Intervention For any number of terms, check for: greatest common factor Techniques for Factoring Polynomials For two terms, check for: Difference of two squares a
More information8-1: Adding and Subtracting Polynomials
8-1: Adding and Subtracting Polynomials Objective: To classify, add, and subtract polynomials Warm Up: Simplify each expression. 1. x 3 7 x 9. 6(3x 4) 3. 7 ( x 8) 4 4. 5(4x (8x 6) monomial - A real number,
More informationFactoring - Factoring Special Products
6.5 Factoring - Factoring Special Products Objective: Identify and factor special products including a difference of squares, perfect squares, and sum and difference of cubes. When factoring there are
More informationreview To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17
1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient
More informationSolving Quadratic Equations Review
Math III Unit 2: Polynomials Notes 2-1 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic
More informationTo solve a radical equation, you must take both sides of an equation to a power.
Topic 5 1 Radical Equations A radical equation is an equation with at least one radical expression. There are four types we will cover: x 35 3 4x x 1x 7 3 3 3 x 5 x 1 To solve a radical equation, you must
More informationUNIT 9 (Chapter 7 BI) Polynomials and Factoring Name:
UNIT 9 (Chapter 7 BI) Polynomials and Factoring Name: The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay
More informationTo factor an expression means to write it as a product of factors instead of a sum of terms. The expression 3x
Factoring trinomials In general, we are factoring ax + bx + c where a, b, and c are real numbers. To factor an expression means to write it as a product of factors instead of a sum of terms. The expression
More informationMath 2142 Homework 5 Part 1 Solutions
Math 2142 Homework 5 Part 1 Solutions Problem 1. For the following homogeneous second order differential equations, give the general solution and the particular solution satisfying the given initial conditions.
More informationUnit 5 Quadratic Expressions and Equations
Unit 5 Quadratic Expressions and Equations Test Date: Name: By the end of this unit, you will be able to Add, subtract, and multiply polynomials Solve equations involving the products of monomials and
More informationSomething that can have different values at different times. A variable is usually represented by a letter in algebraic expressions.
Lesson Objectives: Students will be able to define, recognize and use the following terms in the context of polynomials: o Constant o Variable o Monomial o Binomial o Trinomial o Polynomial o Numerical
More informationQuadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents
Slide 1 / 200 Quadratic Functions Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations
More informationQuadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.
Slide 1 / 200 Quadratic Functions Slide 2 / 200 Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic
More information