Mth 95 Module 2 Spring 2014


 Ophelia Grant
 1 years ago
 Views:
Transcription
1 Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression are separated by addition and subtraction signs. Examples: A monomial is a term, which has only nonnegative integer exponents. (No addition or subtraction signs no division by variables) Monomials Not Monomials The degree of a monomial is the sum of the exponents of its variables. A constant has degree 0, unless the term is 0, which has undefined degree. Give the degree of each monomial. 3 x 17 3 x yz 0 A coefficient is the numerical constant (including its sign) in a monomial. Give the coefficient in each monomial 3 4 4x cd 8 A polynomial is a monomial or a sum of monomials. Polynomials Not Polynomials A binomial is a polynomial which has terms Examples: A trinomial is a polynomial that has terms. Examples: The degree of a polynomial equals the degree of the monomial with the highest degree. 4xy x 3 y 4 xy 6x 3 y + y 5 Degree of each term Polynomial s degree Chapter 5 1
2 Mth 95 Module Spring 014 The leading coefficient of a polynomial of one variable is the coefficient of the monomial with the highest degree Degree Type Example Leading Coefficient 0 Constant 14 1 Linear 3x 1 Quadratic 3y + 5y 4 3 Cubic x 3 4x x + 1 Standard form of a single variable polynomial is written in descending order of the powers of the variable. 3 3x6 7x 4n 5n 6 9n Standard form Degree Leading Term Leading Coefficient Number of Terms To add two polynomials combine like terms. Like terms contain the same variables raised to the same power. (So, x and x are NOT like terms.) Horizontal method Vertical method (3x 4x + 3) + (5x + 4x + 1) Try (3n + n 3 + 4) + (3n 3 8) To subtract two polynomials, add the first polynomial and the opposite of the second polynomial. The opposite of a polynomial is found by changing the sign of every term. (Remember, subtraction is NOT commutative.) (3x + 9) (x + 4) (x + 6) (x + 4x 1) Polynomial functions create smooth, continuous graphs. Remember: Since these are functions, they pass the vertical line test, i.e. for each input there is a unique output. Scale on the axes below is one. The horizontal axis is x; the vertical axis is f(x). x Type: f(x) f(x) = 3 f(x) = x 1 f(x) = x Chapter 5
3 Mth 95 Module Spring 014 f(x) = x 3 other Type: Symbolical Evaluating Polynomial Functions Evaluate f(x) = 4 x Evaluate f(x) = 3x 3 + x 4 at x = 1 and x = 3 at x = and x = 3 Which of the following represent polynomial functions? f(x) = x + 1 f(x) = x 3 f(x) = ½ + 3x x 5 f(x) = 4 x Multiplication of Polynomials Remember distribution 3(x 4) = (x 5y) = Properties of Exponents: For any nonzero numbers a and b and integers m and n, a m a n = a m + n (a m ) n = a mn (ab) n = a n b n Simplifying Polynomials 6x 4 4x 3 = (x y 3 )(3xy ) = 4m (m m 3 ) = xy(x 3 y) = (n 3 m) 3 = (xy ) 4 = 4y(y + 3y 5) = y 3 (y y + 6) = Multiplying Binomials Geometric area model Distribution (x + )(x + 1) (x + )(x + 1) = x (x + 1) + (x + 1) = x + x + x + = Chapter 5 3
4 Mth 95 Module Spring 014 Table Vertical FOIL a shortcut to multiply every term in x + the first binomial by every term in the x + 1 second binomial. (x + )(x + 1) Multiply the FIRST terms x x = x Multiply the OUTER terms x 1 = x Multiply the INNER terms x = x Multiply the LAST terms 1 = Write as a sum x + 3x + (3x )(x + 4) (5 x)(3 5x) (y 1)(y 3) x 7x 11 (n 5) (x 5)(x + 5) Multiplying other polynomials (x + 5)(x 4x + 5) Horizontal method Vertical method (3x )(x + 5x 4) x(x 4x + 5) + 5(x 4x + 5) x 4x + 5 x + 5 Special Products Multiplying binomials that are the sum and difference of the same two terms. ab ab a ab ab b ( difference of squares) y11 y 11 x3x 3 x4x 4 3x3x 1 1 x x (w + 5v )(w 5v ) Chapter 5 4
5 Mth 95 Module Spring 014 Squaring a binomial: Squaring a binomial sum: a b a ba b x 4 (3x + ) What if it was squaring a difference instead of a sum? a b a ba b x 3 x 3 Review: Simplify. Begin by multiplying to change each product to a sum x 3x x 3y 5x y 5y x x x Section 5.5 The Greatest Common Factor and Factoring by Grouping Factoring polynomials is important because we will use factoring to solve polynomial equations. Factoring is undistributing or unfoiling. It is the process of writing a polynomial as the product of polynomials. Distribution x (3x + 5) = 6x + 10x Foiling (x + 3) (x + ) = x + 5x + 6 Factoring 6x + 10x = x (3x + 5) Factoring x + 5x + 6 = (x + 3) (x + ) Factoring out a monomial greatest common factor Always look for the GCF before using other factoring patterns. 10x 3 6 x What factors are common to both terms? 5xxx 3xx Factor each monomial and find the GCF, the highest degree ( x x)(5x) (xx)(3) of each variable in common and the largest common coefficient. x (5x 3) Write as a product of the greatest common monomial and a polynomial Check your factoring by distribution Find the greatest common factor (GCF) y 7, 4y , 4, 8x 4 y 11x 3 y z 4, 33x y 5 z 6 Chapter 5 5
6 Mth 95 Module Spring 014 Factor the following to change each polynomial from a sum to a product. 6x 3 y + 9xy 1x 4 8x 8m 3 m + 6m (When the lead coefficient is negative, factor out a negative GCF.) 40x y + 15 x y x y 3 7xy x 3 6x + 4x Factoring out a binomial greatest common factor This looks a little messier: (x + 3) + x(x + 3) What is the common factor? ( + x) (x + 3) Write as a product. Factoring out a binomial greatest common factor 3 x 7 5y x 7 6x x 3y 5 x 3y x x 3x 3 x x x x 43x x 4 3x x x 4w w 3 w 3 6x x 3y x 3y Factoring by Grouping (Use this if you have four terms after you ve checked for a GCF of all four terms) 1. Group terms in pairs that have a common factor. Factor out the common monomial factor from each pair 3. Factor out, if it exists, the remaining common binomial factor 4. Check factors by multiplying xy 5x 9y x + 6x + 5x + 10 x x 5x Chapter 5 6
7 Mth 95 Module Spring 014 ab a 5b 10 15xy 0x 6y x xy x y xy 4y 3x x x xy y 3 x x x 3 6 Review: Multiply to change each product to a sum. Simplify x x 5 x 3y x 3y x x x 3y 4 Factor each polynomial using GCF or factoring by grouping x y +10x y 6xy 9x 8y 1 Section 5.6 Factoring Trinomials Remember factoring is undoing distribution. It is writing the sum as a product. Leading Coefficient is One: To factor x + bx + c, find integers m and n that satisfy m n = c and m + n = b, then x + bx + c = ( x + m)(x + n). For x + 6x + 8 ask, What pair of integer factors for 8 add to 6? Possible factor pairs are: 1 and 8, and 4, 1 and 8, and and 4 Sums: = = = = 6 Choose and 4 for m and n. Therefore, x + 6x + 8 = (x + )(x + 4). Check by foiling. If a polynomial cannot be factored, we say it is a prime polynomial. If c is positive, m and n are either both positive or both negative. If c is negative, m and n have opposite signs. x 8x 15 x 10x 4 a 8a 16 x 3x 8 y + 8y + 18 x xy 8y Chapter 5 7
8 Mth 95 Module Spring 014 Leading Coefficient is NOT One Sometimes a common factor must be factored out first before factoring the trinomial. Factor completely. 7x x + 4x m 4 + 6m 3 + 5m 3m 3 1m + 36m You can factor ax + bx + c, where a, b, and c have no common factor by the ac method or guess and check. The ac method: 1. Find numbers m and n such that mn = ac and m + n = b.. Write the trinomial as ax + mx + nx + c. 3. Use grouping to factor this expression as two binomials. Example x 5x 3 Since (3) = 6, ask what factors of 6 have a sum of 5. Since 6(1) = 6 and = 5 rewrite 5x as 6x + x x 6x + x 3 Next factor by grouping x(x 3) + 1 (x 3) (x + 1)(x 3) 10y + 13y 3 4n + 19n x 8x x 7x 6 y 5y 3 3x 13x 4 15x 39x 18 1x 5x 14x xy 3y Chapter 5 8
9 Mth 95 Module Spring 014 Section Factoring by Special Products Difference of Two Squares: For any real numbers a and b, a b = (a + b)(a b) Steps in identifying and factoring a difference of squares 1) Is it a binomial (two terms) difference (subtraction)? ) Are both terms perfect squares? (eg. 1, 4, 9, 16,,x, x 4,, 5x, ) 3) Take the square root of each term and substitute into the above pattern. 4) Check by foiling x 9 5x 16 4x + 9 9x 100 (x ) (n 4) y 144 9x 4y Binomials can only be factored by GCF and Difference of Squares. Always check for a GCF before using other factoring patterns. 6x 6y 3x 3 1xy 36x 4y 64z 4 49z Sometimes more than one factoring pattern must be applied or a pattern must be applied more than once. x 4 y 4 x 3 + x 9x 9 x 81y x 8x x Chapter 5 9
10 Mth 95 Module Spring 014 Perfect Square Trinomials: For and real numbers a and b, a + ab + b = (a + b) and a  ab + b = (a  b) Steps in identifying and factoring a difference of squares 1) Is it a trinomial (three terms) with the powers of the variable in order? ) Are the first and third terms POSITIVE perfect squares? 3) Multiply the square roots of these two terms and double the result. If it is a perfect square trinomial, the answer should be the same as or the opposite of the middle term of the polynomial. 4) Factor using one of the patterns above. 5) Check by foiling. x + x + 1 x 6x + 9 9y 6y 4 5x + 30xy + 9y 4z 4z 1 x x 1 x 0x x 16x 1 A General Factoring Strategy 1. For two or more terms, factor out any common factor..if you have only two terms, check for difference of squares. 3.If you have three terms, factor by trial and error, use the ac method or use one of the perfect square trinomial patterns. 4.If you have four terms, factor by grouping. 5.Continue factoring until none of the polynomial factors can be factored further. Factoring Flow Chart GCF Two terms: Use Difference of Squares a b = (a + b)(a b) Three terms: Use x + bx +c ax + bx + c, where a 1 a + ab + b = (a + b) a  ab + b = (a  b) Four terms: Use Factoring by grouping Chapter 5 10
11 Mth 95 Module Spring 014 Factor completely. 3x y 75y 3 x y 16y + 3 x 3y + 4y 6 4z 17z 4 4x 3x 60 18x + 30x x +5x 1 4x x 10 75a 48 x 5 3 n n n r 1r t 18rt 3 5x x 7 3xy 6y 5x 10 4r t 4 6 Chapter 5 11
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.
More informationx n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent.
Rules of Exponents: If n > 0, m > 0 are positive integers and x, y are any real numbers, then: x m x n = x m+n x m x n = xm n, if m n (x m ) n = x mn (xy) n = x n y n ( x y ) n = xn y n 1 Can we make sense
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More informationUnit 3 Polynomials Study Guide
Unit Polynomials Study Guide 75 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial
More informationName Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
More informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationSection 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
More informationAlgebra 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
More informationFACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1
5.7 Factoring ax 2 bx c (549) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationChapter 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
More informationMonomials with the same variables to the same powers are called like terms, If monomials are like terms only their coefficients can differ.
Chapter 7.1 Introduction to Polynomials A monomial is an expression that is a number, a variable or the product of a number and one or more variables with nonnegative exponents. Monomials that are real
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationSTUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS
STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an
More informationOperations 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
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationPOLYNOMIALS and FACTORING
POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use
More informationUnit: Polynomials and Factoring
Name Unit: Polynomials: Multiplying and Factoring Specific Outcome 10I.A.1 Demonstrate an understanding of factors of whole numbers by determining: Prime factors Greatest common factor Least common multiple
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationFACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c
Tallahassee Community College 55 FACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c This kind of trinomial differs from the previous kind we have factored because the coefficient of x is no longer "1".
More informationFactoring. Factoring Monomials Monomials can often be factored in more than one way.
Factoring Factoring is the reverse of multiplying. When we multiplied monomials or polynomials together, we got a new monomial or a string of monomials that were added (or subtracted) together. For example,
More informationFactoring 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
More informationFactoring Flow Chart
Factoring Flow Chart greatest common factor? YES NO factor out GCF leaving GCF(quotient) how many terms? 4+ factor by grouping 2 3 difference of squares? perfect square trinomial? YES YES NO NO a 2 b
More informationSummer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2
Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level
More informationTool 1. Greatest Common Factor (GCF)
Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
More informationAlgebra 1 Chapter 3 Vocabulary. equivalent  Equations with the same solutions as the original equation are called.
Chapter 3 Vocabulary equivalent  Equations with the same solutions as the original equation are called. formula  An algebraic equation that relates two or more reallife quantities. unit rate  A rate
More informationMATH 90 CHAPTER 6 Name:.
MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a
More informationChapter 4. Polynomials
4.1. Add and Subtract Polynomials KYOTE Standards: CR 8; CA 2 Chapter 4. Polynomials Polynomials in one variable are algebraic expressions such as 3x 2 7x 4. In this example, the polynomial consists of
More informationVocabulary 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 informationGreatest 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
More informationThe xintercepts of the graph are the xvalues for the points where the graph intersects the xaxis. A parabola may have one, two, or no xintercepts.
Chapter 101 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax
More informationChapter 3. Algebra. 3.1 Rational expressions BAa1: Reduce to lowest terms
Contents 3 Algebra 3 3.1 Rational expressions................................ 3 3.1.1 BAa1: Reduce to lowest terms...................... 3 3.1. BAa: Add, subtract, multiply, and divide............... 5
More informationDefinitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).
Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32
More informationThis is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).
This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationIn algebra, factor by rewriting a polynomial as a product of lowerdegree 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
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationName Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE
Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers
More informationA. Factoring out the Greatest Common Factor.
DETAILED SOLUTIONS AND CONCEPTS  FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
More informationPolynomials and Factoring
7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of
More information6.1 The Greatest Common Factor; Factoring by Grouping
386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.
More informationA Systematic Approach to Factoring
A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool
More informationPolynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
More informationFactoring Algebra Chapter 8B Assignment Sheet
Name: Factoring Algebra Chapter 8B Assignment Sheet Date Section Learning Targets Assignment Tues 2/17 Find the prime factorization of an integer Find the greatest common factor (GCF) for a set of monomials.
More informationQuadratic Equations and Inequalities
MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationBy reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.
SECTION 5.4 Special Factoring Techniques 317 5.4 Special Factoring Techniques OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor
More information3. Power of a Product: Separate letters, distribute to the exponents and the bases
Chapter 5 : Polynomials and Polynomial Functions 5.1 Properties of Exponents Rules: 1. Product of Powers: Add the exponents, base stays the same 2. Power of Power: Multiply exponents, bases stay the same
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationWhen factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.
Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property
More informationFactoring 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
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationUsing the ac Method to Factor
4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trialanderror
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More informationFactoring Polynomials
Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,
More informationFACTORING POLYNOMIALS
296 (540) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated
More informationMath PreCalc 20 Chapter 4 Review of Factoring. Questions to try. 2. x 2 6xy x x x x 2 y + 8xy
Math PreCalc 20 Chapter 4 Review of Factoring Multiplying (Expanding) Type 1: Monomial x Binomial Monomial x Trinomial Ex: 3(x + 4) = 3x + 122(x 2 + 2x 1) = 2x 2 4x + 2 Multiply the following: 1. 5(x
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationLagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.
Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method
More informationSIMPLIFYING ALGEBRAIC FRACTIONS
Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is
More informationPolynomials. 44 to 48
Polynomials 44 to 48 Learning Objectives 44 Polynomials Monomials, binomials, and trinomials Degree of a polynomials Evaluating polynomials functions Polynomials Polynomials are sums of these "variables
More informationThe Greatest Common Factor; Factoring by Grouping
296 CHAPTER 5 Factoring and Applications 5.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.
More informationCOGNITIVE TUTOR ALGEBRA
COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,
More informationMath 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 informationFactor Polynomials Completely
9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping
More informationUnit 1: Polynomials. Expressions:  mathematical sentences with no equal sign. Example: 3x + 2
Pure Math 0 Notes Unit : Polynomials Unit : Polynomials : Reviewing Polynomials Epressions:  mathematical sentences with no equal sign. Eample: Equations:  mathematical sentences that are equated with
More information15.1 Factoring Polynomials
LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE
More informationFactoring Trinomials of the Form x 2 bx c
4.2 Factoring Trinomials of the Form x 2 bx c 4.2 OBJECTIVES 1. Factor a trinomial of the form x 2 bx c 2. Factor a trinomial containing a common factor NOTE The process used to factor here is frequently
More informationDifference of Squares and Perfect Square Trinomials
4.4 Difference of Squares and Perfect Square Trinomials 4.4 OBJECTIVES 1. Factor a binomial that is the difference of two squares 2. Factor a perfect square trinomial In Section 3.5, we introduced some
More informationFactoring (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
More informationPreCalculus III Linear Functions and Quadratic Functions
Linear Functions.. 1 Finding Slope...1 Slope Intercept 1 Point Slope Form.1 Parallel Lines.. Line Parallel to a Given Line.. Perpendicular Lines. Line Perpendicular to a Given Line 3 Quadratic Equations.3
More informationMonomial. 5 1 x A sum is not a monomial. 2 A monomial cannot have a. x 21. degree. 2x 3 1 x 2 2 5x Rewrite a polynomial
9.1 Add and Subtract Polynomials Before You added and subtracted integers. Now You will add and subtract polynomials. Why? So you can model trends in recreation, as in Ex. 37. Key Vocabulary monomial degree
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationFactoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1)
Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if
More informationALGEBRA 2: 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 informationPolynomials. Solving Equations by Using the Zero Product Rule
mil23264_ch05_303396 9:21:05 06:16 PM Page 303 Polynomials 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials 5.4 Greatest
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationAlgebra Tiles Activity 1: Adding Integers
Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting
More informationMyMathLab ecourse for Developmental Mathematics
MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and
More informationFactoring Special Polynomials
6.6 Factoring Special Polynomials 6.6 OBJECTIVES 1. Factor the difference of two squares 2. Factor the sum or difference of two cubes In this section, we will look at several special polynomials. These
More informationMath 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 information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationSection 2.1 Intercepts; Symmetry; Graphing Key Equations
Intercepts: An intercept is the point at which a graph crosses or touches the coordinate axes. x intercept is 1. The point where the line crosses (or intercepts) the xaxis. 2. The xcoordinate of a point
More information78 Multiplying Polynomials
78 Multiplying Polynomials California Standards 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these
More informationState whether each sentence is true or false. If false, replace the underlined phrase or expression to make a true sentence.
State whether each sentence is true or false. If false, replace the underlined phrase or expression to make a true sentence. 1. x + 5x + 6 is an example of a prime polynomial. The statement is false. A
More informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More informationIntroduction Assignment
PRECALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying
More informationPolynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c
More informationAlgebra Unit 6 Syllabus revised 2/27/13 Exponents and Polynomials
Algebra Unit 6 Syllabus revised /7/13 1 Objective: Multiply monomials. Simplify expressions involving powers of monomials. Preassessment: Exponents, Fractions, and Polynomial Expressions Lesson: Pages
More informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationMultiplying Polynomials 5
Name: Date: Start Time : End Time : Multiplying Polynomials 5 (WS#A10436) Polynomials are expressions that consist of two or more monomials. Polynomials can be multiplied together using the distributive
More information