Polynomials and Factoring
|
|
- Harold Paul
- 6 years ago
- Views:
Transcription
1 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 terms, with only nonnegative integer eponents permitted on the variables. If the terms of a polynomial contain only the variable, then the polynomial is called a polynomial in. (Polynomials in other variables are defined similarly.) Eamples of polynomials include , 9p 5 3, 8r 2, and 6. The epression is not a polynomial because of the presence of 6. The terms of a polynomial cannot have variables in a denominator. The greatest eponent in a polynomial in one variable is the degree of the polynomial. A nonzero constant is said to have degree 0. (The polynomial 0 has no degree.) For eample, is a polynomial of degree 6. A polynomial can have more than one variable. A term containing more than one variable has degree equal to the sum of all the eponents appearing on the variables in the term. For eample, 3 4 y 3 z 5 is of degree The degree of a polynomial in more than one variable is equal to the greatest degree of any term appearing in the polynomial. By this definition, the polynomial is of degree 8 because of the 6 y 2 term. 2 4 y y 6 y 2
2 7.6 Polynomials and Factoring 373 A polynomial containing eactly three terms is called a trinomial and one containing eactly two terms is a binomial. For eample, is a trinomial of degree 9. The table shows several polynomials and gives the degree and type of each. Polynomial Degree Type 9p 7 4p 3 8p r 6 s 8 5a 3 b 7 3a 5 b 5 4a 2 b 9 a 10 7 Trinomial 15 Binomial 14 Monomial 11 None of these Addition and Subtraction Since the variables used in polynomials represent real numbers, a polynomial represents a real number. This means that all the properties of the real numbers mentioned in this book hold for polynomials. In particular, the distributive property holds, so 3m 5 7m 5 3 7m 5 4m 5. Like terms are terms that have the eact same variable factors. Thus, polynomials are added by adding coefficients of like terms; polynomials are subtracted by subtracting coefficients of like terms. EXAMPLE 1 Add or subtract, as indicated. 2y 4 3y 2 y 4y 4 7y 2 6y 2 4y 4 3 7y 2 1 6y 6y 4 4y 2 7y 3m 3 8m 2 4 m 3 7m m 3 8 7m m 3 15m 2 7 (c) 8m 4 p 5 9m 3 p 5 11m 4 p 5 15m 3 p 5 19m 4 p 5 6m 3 p 5 (d) Distributive property Associative property Add like terms. As shown in parts,, and (d) of Eample 1, polynomials in one variable are often written with their terms in descending powers; so the term of greatest degree is first, the one with the net greatest degree is second, and so on. Multiplication The associative and distributive properties, together with the properties of eponents, can also be used to find the product of two polynomials. For eample, to find the product of 3 4 and , treat 3 4 as a single epression and use the distributive property as follows
3 374 CHAPTER 7 The Basic Concepts of Algebra Now use the distributive property three separate times on the right of the equality symbol to get It is sometimes more convenient to write such a product vertically, as follows k k Add in columns. EXAMPLE 2 Multiply 3p 2 4p 1p 3 2p 8. Multiply each term of the second polynomial by each term of the first and add these products. It is most efficient to work vertically with polynomials of more than two terms, so that like terms can be placed in columns. 3p 2 4p 1 p 3 2p 8 24p 2 32p 8 Multiply 3p 2 4p 1 by 8. 6p 3 8p 2 2p Multiply 3p 2 4p 1 by 2p. 3p 5 4p 4 p 3 Multiply 3p 2 4p 1 by p 3. 3p 5 4p 4 7p 3 32p 2 34p 8 Add in columns. The FOIL method is a convenient way to find the product of two binomials. The memory aid FOIL (for First, Outside, Inside, Last) gives the pairs of terms to be multiplied to get the product, as shown in the net eamples. The special product y y 2 y 2 can be used to solve some multiplication problems. For eample, , Once these patterns are recognized, multiplications of this type can be done mentally. EXAMPLE 3 Find each product. F O I L 6m 14m 3 6m4m 6m3 14m 13 24m 2 14m In part of Eample 3, the product of two binomials was a trinomial, while in part the product of two binomials was a binomial. The product of two binomials of the forms y and y is always a binomial. Check by multiplying that the following is true. Product of the Sum and Difference of Two Terms y y 2 y 2
4 7.6 Polynomials and Factoring 375 This product is called the difference of two squares. Since products of this type occur frequently, it is important to be able to recognize when this pattern should be used. EXAMPLE 4 Find each product. 3p 113p 11 Using the pattern discussed above, replace with 3p and y with 11. (c) 3p 113p 11 3p p m 3 35m 3 3 5m m 6 9 9k 11r 3 9k 11r 3 9k 2 11r k 2 121r 6 The squares of binomials are also special products. Squares of Binomials y 2 2 2y y 2 y 2 2 2y y 2 y Area: 2 Area: y The special product y 2 2 2y y 2 can be illustrated geometrically using the diagram shown here. Each side of the large square has length y, so the area of the square is y 2. The large square is made up of two smaller squares and two congruent rectangles. The sum of the areas of these figures is 2 2y y 2. Area: y Area: y 2 Since these epressions represent the same quantity, they must be equal, thus giving us the pattern for squaring a binomial. y EXAMPLE 5 Find each product. 2m 5 2 2m 2 22m m 2 20m y y 4 7y y 4 49y 8 As shown in Eample 5, the square of a binomial has three terms. Students often mistakenly give 2 y 2 as equivalent to the product y 2. Be careful to avoid that error. The process of finding polynomials whose product equals a given polynomial is called factoring. For eample, since , both 4 and 3 are called factors of Also, 4 3 is called the factored form of A polynomial that cannot be written as a product of two polynomials with integer coefficients is a prime polynomial. A polynomial is factored completely when it is written as a product of prime polynomials with integer coefficients. Factoring Out the Greatest Common Factor Some polynomials are factored by using the distributive property. For eample, to factor 6 2 y 3 9y 4 18y 5, we look for a monomial that is the greatest common factor of all the terms of the polynomial. For this polynomial, 3y 3 is the greatest common factor. By the distributive property, 6 2 y 3 9y 4 18y 5 3y y 3 3y 3y 3 6y 2 3y y 6y 2.
5 376 CHAPTER 7 The Basic Concepts of Algebra EXAMPLE 6 Factor out the greatest common factor from each polynomial. 9y 5 y 2 y 2 9y 3 y 2 1 The greatest common factor is y 2. y 2 9y 3 1 (c) 6 2 t 8t 12t 2t m 4 m 1 28m 3 m 1 7m 2 m 1 The greatest common factor is 7m 2 m 1. Use the distributive property. 14m 4 m 1 28m 3 m 1 7m 2 m 1 7m 2 m 12m 2 4m 1 7m 2 m 12m 2 4m 1 Factoring by Grouping When a polynomial has more than three terms, it can sometimes be factored by a method called factoring by grouping. For eample, to factor collect the terms into two groups so that each group has a common factor. Factor each group, getting a ay 6 6y, a ay 6 6y a ay 6 6y a ay 6 6y a y 6 y. The quantity y is now a common factor, which can be factored out, producing a ay 6 6y ya 6. It is not always obvious which terms should be grouped. Eperience and repeated trials are the most reliable tools for factoring by grouping. EXAMPLE 7 Factor by grouping. mp 2 7m 3p 2 21 mp 2 7m 3p 2 21 m p p 2 7 p 2 7m 3 Group the terms. Factor each group. p 2 7 is a common factor. 2y 2 2z ay 2 az 2y 2 2z ay 2 az 2y 2 z ay 2 z Factor each group. The epression y 2 z is the negative of y 2 z, so factor out a instead of a. Factoring Trinomials 2y 2 z ay 2 z y 2 z2 a Factor out a. Factor out y 2 z. Factoring is the opposite of multiplying. Since the product of two binomials is usually a trinomial, we can epect factorable trinomials (that have terms with no common factor) to have two binomial factors. Thus, factoring trinomials requires using FOIL backward.
6 7.6 Polynomials and Factoring 377 EXAMPLE 8 Factor each trinomial. 4y 2 11y 6 To factor this polynomial, we must find integers a, b, c, and d such that 4y 2 11y 6 ay bcy d. By using FOIL, we see that ac 4 and bd 6. The positive factors of 4 are 4 and 1 or 2 and 2. Since the middle term is negative, we consider only negative factors of 6. The possibilities are 2 and 3 or 1 and 6. Now we try various arrangements of these factors until we find one that gives the correct coefficient of y. 2y 12y 6 4y 2 14y 6 2y 22y 3 4y 2 10y 6 y 24y 3 4y 2 11y 6 Incorrect Incorrect Correct The last trial gives the correct factorization. 6p 2 7p 5 Again, we try various possibilities. The positive factors of 6 could be 2 and 3 or 1 and 6. As factors of 5 we have only 1 and 5 or 5 and 1. Try different combinations of these factors until the correct one is found. 2p 53p 1 6p 2 13p 5 3p 52p 1 6p 2 7p 5 Incorrect Correct Thus, 6p 2 7p 5 factors as 3p 52p 1. Each of the special patterns of multiplication given earlier can be used in reverse to get a pattern for factoring. Perfect square trinomials can be factored as follows. Perfect Square Trinomials 2 2y y 2 y 2 2 2y y 2 y 2 EXAMPLE 9 Factor each polynomial. 16p 2 40pq 25q 2 Since 16p 2 4p 2 and 25q 2 5q 2, use the second pattern shown above with 4p replacing and 5q replacing y to obtain 16p 2 40pq 25q 2 4p 2 24p5q 5q 2 4p 5q 2. Make sure that the middle term of the trinomial being factored, 40pq here, is twice the product of the two terms in the binomial 4p 5q. 40pq 24p5q y 2 16y y 2 2, since 2134y 2 104y 2.
7 378 CHAPTER 7 The Basic Concepts of Algebra y y y y Factoring Binomials The pattern for the product of the sum and difference of two terms gives the following factorization. Difference of Squares 2 y 2 y y A geometric proof for the difference of squares property is shown above. (The proof is only valid for y 0.) 2 y 2 y y y y y Factor out y in the second step. EXAMPLE 10 Factor each of the following polynomials. 4m 2 9 First, recognize that 4m 2 9 is the difference of squares, since 4m 2 2m 2 and Use the pattern for the difference of squares with 2m replacing and 3 replacing y. Doing this gives 4m 2 9 2m m 32m k 4 625m 4 Use the difference of squares pattern twice. 256k 4 625m 4 16k m k 2 25m 2 16k 2 25m 2 16k2 25m2 4k 5m4k 5m (c) y 4 Group the first three terms to obtain a perfect square trinomial. Then use the difference of squares pattern y y y y 2 3 y 2 3 y 2 3 y 2 Two other special results of factoring are listed below. Each can be verified by multiplying on the right side of the equation. Sum and Difference of Cubes Sum of Cubes Difference of Cubes 3 y 3 y 2 y y 2 3 y 3 y 2 y y 2 EXAMPLE 11 Factor each polynomial Notice that , so the epression is a sum of cubes. Use the first pattern given above
8 7.6 Polynomials and Factoring 379 m 3 64n 3 m 3 4n 3 m 4nm 2 m4n 4n 2 m 4nm 2 4mn 16n 2 (c) 8q 6 125p 9 2q 2 3 5p 3 3 2q 2 5p 3 2q 2 2 2q 2 5p 3 5p 3 2 2q 2 5p 3 4q 4 10q 2 p 3 25p 6
Algebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.
C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each
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 informationMath 154 :: Elementary Algebra
Math :: Elementar Algebra Section. Section. Section. Section. Section. Section. Math :: Elementar Algebra Section. Eponents. When multipling like-bases, ou can add the eponents to simplif the epression..
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 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 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 information1.3 Algebraic Expressions. Copyright Cengage Learning. All rights reserved.
1.3 Algebraic Expressions Copyright Cengage Learning. All rights reserved. Objectives Adding and Subtracting Polynomials Multiplying Algebraic Expressions Special Product Formulas Factoring Common Factors
More informationAlgebra I Notes Unit Eleven: Polynomials
Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. Teacher Note: A nice way to illustrate operations with polynomials
More informationLesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o
Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o o ( 1)(9) 3 ( 1) 3 9 1 Evaluate the second expression at the left, if
More informationAlgebraic Expressions and Identities
9 Algebraic Epressions and Identities introduction In previous classes, you have studied the fundamental concepts of algebra, algebraic epressions and their addition and subtraction. In this chapter, we
More information7.3 Adding and Subtracting Rational Expressions
7.3 Adding and Subtracting Rational Epressions LEARNING OBJECTIVES. Add and subtract rational epressions with common denominators. 2. Add and subtract rational epressions with unlike denominators. 3. Add
More informationIn this unit we will study exponents, mathematical operations on polynomials, and factoring.
GRADE 0 MATH CLASS NOTES UNIT E ALGEBRA In this unit we will study eponents, mathematical operations on polynomials, and factoring. Much of this will be an etension of your studies from Math 0F. This unit
More informationChapter 5: Exponents and Polynomials
Chapter 5: Exponents and Polynomials 5.1 Multiplication with Exponents and Scientific Notation 5.2 Division with Exponents 5.3 Operations with Monomials 5.4 Addition and Subtraction of Polynomials 5.5
More informationTHE DISTRIBUTIVE LAW. Note: To avoid mistakes, include arrows above or below the terms that are being multiplied.
THE DISTRIBUTIVE LAW ( ) When an equation of the form a b c is epanded, every term inside the bracket is multiplied by the number or pronumeral (letter), and the sign that is located outside the brackets.
More informationHow to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials
PRC Ch P_3.notebook How to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials How to remove common factors from polynomials
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 informationAlgebra Final Exam Review Packet
Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:
More informationMA 22000, Lesson 2 Functions & Addition/Subtraction Polynomials Algebra section of text: Sections 3.5 and 5.2, Calculus section of text: Section R.
MA 000, Lesson Functions & Addition/Subtraction Polynomials Algebra section of tet: Sections.5 and 5., Calculus section of tet: Section R.1 Definition: A relation is any set of ordered pairs. The set of
More informationChapter 6: Polynomials
Chapter : Polynomials Chapter : Polynomials POLYNOMIALS Definition: A polynomial is an algebraic epression that is a sum of terms, where each term contains only variables with whole number eponents and
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 informationDivisibility Rules Algebra 9.0
Name Period Divisibility Rules Algebra 9.0 A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following eercise: 1. Cross
More informationPre-Algebra Notes Unit 12: Polynomials and Sequences
Pre-Algebra Notes Unit 1: Polynomials and Sequences Polynomials Syllabus Objective: (6.1) The student will write polynomials in standard form. Let s review a definition: monomial. A monomial is a number,
More informationBishop Kelley High School Summer Math Program Course: Algebra 2 A
06 07 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 6 pages of this packet provide eamples as to how to work some of the problems
More informationEby, MATH 0310 Spring 2017 Page 53. Parentheses are IMPORTANT!! Exponents only change what they! So if a is not inside parentheses, then it
Eby, MATH 010 Spring 017 Page 5 5.1 Eponents Parentheses are IMPORTANT!! Eponents only change what they! So if a is not inside parentheses, then it get raised to the power! Eample 1 4 b) 4 c) 4 ( ) d)
More informationI CAN classify polynomials by degree and by the number of terms.
13-1 Polynomials I CAN classify polynomials by degree and by the number of terms. 13-1 Polynomials Insert Lesson Title Here Vocabulary monomial polynomial binomial trinomial degree of a polynomial 13-1
More information5 3w. Unit 2 Function Operations and Equivalence Standard 4.1 Add, Subtract, & Multiply Polynomials
Unit Function Operations and Equivalence This document is meant to be used as an eample guide for each of the skills we will be holding students accountable for with Standard 4.1. This document should
More informationMini Lecture 9.1 Finding Roots
Mini Lecture 9. Finding Roots. Find square roots.. Evaluate models containing square roots.. Use a calculator to find decimal approimations for irrational square roots. 4. Find higher roots. Evaluat. a.
More informationSections 7.2, 7.3, 4.1
Sections 7., 7.3, 4.1 Section 7. Multiplying, Dividing and Simplifying Radicals This section will discuss the rules for multiplying, dividing and simplifying radicals. Product Rule for multiplying radicals
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 informationa = B. Examples: 1. Simplify the following expressions using the multiplication rule
Section. Monomials Objectives:. Multiply and divide monomials.. Simplify epressions involving powers of monomials.. Use epressions in scientific notation. I. Negative Eponents and Eponents of Zero A. Rules.
More informationCan that be Axl, your author s yellow lab, sharing a special
46 Chapter P Prerequisites: Fundamental Concepts Algebra Objectives Section Understand the vocabulary polynomials. Add and subtract polynomials. Multiply polynomials. Use FOIL in polynomial multiplication.
More informationEXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n
Algebra B: Chapter 6 Notes 1 EXPONENT REVIEW!!! Concept Byte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Property of Eponents: Product of Powers m n = m
More informationTECHNIQUES IN FACTORISATION
TECHNIQUES IN FACTORISATION The process where brackets are inserted into an equation is referred to as factorisation. Factorisation is the opposite process to epansion. METHOD: Epansion ( + )( 5) 15 Factorisation
More informationBasic ALGEBRA 2 SUMMER PACKET
Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout
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 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 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 informationUnit 3 Factors & Products
1 Unit 3 Factors & Products General Outcome: Develop algebraic reasoning and number sense. Specific Outcomes: 3.1 Demonstrate an understanding of factors of whole number by determining the: o prime factors
More informationMath 10-C Polynomials Concept Sheets
Math 10-C Polynomials Concept Sheets Concept 1: Polynomial Intro & Review A polynomial is a mathematical expression with one or more terms in which the exponents are whole numbers and the coefficients
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 informationMultiplying Monomials
320 Chapter 5 Polynomials Eample 1 Multiplying Monomials Multiply the monomials. a. 13 2 y 7 215 3 y2 b. 1 3 4 y 3 21 2 6 yz 8 2 a. 13 2 y 7 215 3 y2 13 521 2 3 21y 7 y2 15 5 y 8 Group coefficients and
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 informationReview Unit Multiple Choice Identify the choice that best completes the statement or answers the question.
Review Unit 3 1201 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which of the following numbers is not both a perfect square and a perfect cube? a. 531
More informationUniversity of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra
University of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra Table of Contents Chapter The Algebra of Polynomials Chapter Factoring 7 Chapter 3 Fractions Chapter 4 Eponents and Radicals
More informationA2T. Rational Expressions/Equations. Name: Teacher: Pd:
AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 1-3 HW: Pages -3 in Packet o Day : SWBAT: Factor using the Greatest
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 informationIntermediate Algebra 100A Final Exam Review Fall 2007
1 Basic Concepts 1. Sets and Other Basic Concepts Words/Concepts to Know: roster form, set builder notation, union, intersection, real numbers, natural numbers, whole numbers, integers, rational numbers,
More informationSummer MA Lesson 11 Section 1.5 (part 1)
Summer MA 500 Lesson Section.5 (part ) The general form of a quadratic equation is a + b + c = 0, where a, b, and c are real numbers and a 0. This is a second degree equation. There are four ways to possibly
More informationControlling the Population
Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1
More informationMATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline
MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline 1. Real Numbers (33 topics) 1.3 Fractions (pg. 27: 1-75 odd) A. Simplify fractions. B. Change mixed numbers
More informationLESSON 9.1 ROOTS AND RADICALS
LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical
More informationACCUPLACER MATH 0310
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to
More informationNIT #7 CORE ALGE COMMON IALS
UN NIT #7 ANSWER KEY POLYNOMIALS Lesson #1 Introduction too Polynomials Lesson # Multiplying Polynomials Lesson # Factoring Polynomials Lesson # Factoring Based on Conjugate Pairs Lesson #5 Factoring Trinomials
More informationILLUSTRATIVE EXAMPLES
CHAPTER Points to Remember : POLYNOMIALS 7. A symbol having a fied numerical value is called a constant. For e.g. 9,,, etc.. A symbol which may take different numerical values is known as a variable. We
More informationHarbor Creek School District. Algebra II Advanced. Concepts Timeframe Skills Assessment Standards Linear Equations Inequalities
Algebra II Advanced and Graphing and Solving Linear Linear Absolute Value Relation vs. Standard Forms of Linear Slope Parallel & Perpendicular Lines Scatterplot & Linear Regression Graphing linear Absolute
More informationCh. 7.1 Polynomial Degree & Finite Differences
Ch. 7.1 Polynomial Degree & Finite Differences Learning Intentions: Define terminology associated with polynomials: term, monomial, binomial & trinomial. Use the finite differences method to determine
More informationMaintaining Mathematical Proficiency
Chapter 7 Maintaining Mathematical Proficiency Simplify the expression. 1. 5x 6 + 3x. 3t + 7 3t 4 3. 8s 4 + 4s 6 5s 4. 9m + 3 + m 3 + 5m 5. 4 3p 7 3p 4 1 z 1 + 4 6. ( ) 7. 6( x + ) 4 8. 3( h + 4) 3( h
More informationALGEBRAIC EXPRESSIONS AND POLYNOMIALS
MODULE - ic Epressions and Polynomials ALGEBRAIC EXPRESSIONS AND POLYNOMIALS So far, you had been using arithmetical numbers, which included natural numbers, whole numbers, fractional numbers, etc. and
More informationAlgebra. Robert Taggart
Algebra Robert Taggart Table of Contents To the Student.............................................. v Unit 1: Algebra Basics Lesson 1: Negative and Positive Numbers....................... Lesson 2: Operations
More informationDegree of a polynomial
Variable Algebra Term Polynomial Monomial Binomial Trinomial Degree of a term Degree of a polynomial Linear A generalization of arithmetic. Letters called variables are used to denote numbers, which are
More informationMini-Lecture 5.1 Exponents and Scientific Notation
Mini-Lecture.1 Eponents and Scientific Notation Learning Objectives: 1. Use the product rule for eponents.. Evaluate epressions raised to the zero power.. Use the quotient rule for eponents.. Evaluate
More informationM098 Carson Elementary and Intermediate Algebra 3e Section 11.3
Objectives. Solve equations by writing them in quadratic form.. Solve equations that are quadratic in form by using substitution. Vocabulary Prior Knowledge Solve rational equations: Clear the fraction.
More informationBishop Kelley High School Summer Math Program Course: Algebra 2 A
015 016 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 16 pages of this packet provide eamples as to how to work some of the problems
More informationName Class Date. Multiplying Two Binomials Using Algebra Tiles
Name Class Date Multiplying Polynomials Going Deeper Essential question: How do you multiply polynomials? 6-5 A monomial is a number, a variable, or the product of a number and one or more variables raised
More informationUnit 13: Polynomials and Exponents
Section 13.1: Polynomials Section 13.2: Operations on Polynomials Section 13.3: Properties of Exponents Section 13.4: Multiplication of Polynomials Section 13.5: Applications from Geometry Section 13.6:
More informationSection 6.2 Long Division of Polynomials
Section 6. Long Division of Polynomials INTRODUCTION In Section 6.1 we learned to simplify a rational epression by factoring. For eample, + 3 10 = ( + 5)( ) ( ) = ( + 5) 1 = + 5. However, if we try to
More informationStudy Guide for Math 095
Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.
More informationAlgebra II Notes Polynomial Functions Unit Introduction to Polynomials. Math Background
Introduction to Polynomials Math Background Previously, you Identified the components in an algebraic epression Factored quadratic epressions using special patterns, grouping method and the ac method Worked
More informationTEKS: 2A.10F. Terms. Functions Equations Inequalities Linear Domain Factor
POLYNOMIALS UNIT TEKS: A.10F Terms: Functions Equations Inequalities Linear Domain Factor Polynomials Monomial, Like Terms, binomials, leading coefficient, degree of polynomial, standard form, terms, Parent
More informationCOUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra
COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed
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 informationMath 3 Variable Manipulation Part 3 Polynomials A
Math 3 Variable Manipulation Part 3 Polynomials A 1 MATH 1 & 2 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does
More informationP.1: Algebraic Expressions, Mathematical Models, and Real Numbers
Chapter P Prerequisites: Fundamental Concepts of Algebra Pre-calculus notes Date: P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Algebraic expression: a combination of variables and
More informationPrerequisites. Copyright Cengage Learning. All rights reserved.
Prerequisites P Copyright Cengage Learning. All rights reserved. P.4 FACTORING POLYNOMIALS Copyright Cengage Learning. All rights reserved. What You Should Learn Remove common factors from polynomials.
More informationChapter 8 Class Notes 8-A1 (Lessons 8-1&8-2) Monomials and Factoring p Prime Factorization: a whole number expressed as the of factors.
Chapter 8 Class Notes Alg. 1H 8-A1 (Lessons 8-1&8-) Monomials and Factoring p. 40-4 Prime Factorization: a whole number epressed as the of factors. Tree Method: Ladder Method: Factored Form of a Monomial:
More informationReady To Go On? Skills Intervention 7-1 Integer Exponents
7A Evaluating Expressions with Zero and Negative Exponents Zero Exponent: Any nonzero number raised to the zero power is. 4 0 Ready To Go On? Skills Intervention 7-1 Integer Exponents Negative Exponent:
More informationIntensive Math-Algebra I Mini-Lesson MA.912.A.4.3
Intensive Math-Algebra I Mini-Lesson M912.4.3 Summer 2013 Factoring Polynomials Student Packet Day 15 Name: Date: Benchmark M912.4.3 Factor polynomials expressions This benchmark will be assessed using
More informationClassifying Polynomials. Classifying Polynomials by Numbers of Terms
Lesson -2 Lesson -2 Classifying Polynomials BIG IDEA Polynomials are classifi ed by their number of terms and by their degree. Classifying Polynomials by Numbers of Terms Recall that a term can be a single
More informationSOLVING QUADRATIC EQUATIONS USING GRAPHING TOOLS
GRADE PRE-CALCULUS UNIT A: QUADRATIC EQUATIONS (ALGEBRA) CLASS NOTES. A definition of Algebra: A branch of mathematics which describes basic arithmetic relations using variables.. Algebra is just a language.
More informationLake Elsinore Unified School District Pacing Guide & Benchmark Assessment Schedule Algebra 1 Essentials
1.0 Students identify and use the arithmetic properties of subsets of integers, including closure properties for the four basic arithmetic operations where applicable: 1.1 Students use properties of numbers
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 informationJAMES WOOD HIGH SCHOOL 161 Apple Pie Ridge Road Winchester, VA (540) FAX (540)
JAMES WOOD HIGH SCHOOL 161 Apple Pie Ridge Road Winchester, VA 603-4118 (540) 667-56 FAX (540) 667-3154 Summer Math Packet Ms. K. Hill hillk@fcpsk1.net Honors Algebra Name N School o Welcome to Honors
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 informationRadical Expressions and Functions What is a square root of 25? How many square roots does 25 have? Do the following square roots exist?
Topic 4 1 Radical Epressions and Functions What is a square root of 25? How many square roots does 25 have? Do the following square roots eist? 4 4 Definition: X is a square root of a if X² = a. 0 Symbolically,
More informationGraphs of Polynomials: Polynomial functions of degree 2 or higher are smooth and continuous. (No sharp corners or breaks).
Graphs of Polynomials: Polynomial functions of degree or higher are smooth and continuous. (No sharp corners or breaks). These are graphs of polynomials. These are NOT graphs of polynomials There is a
More informationa b + c b = a+c a b c d = ac a b c d = a b d a does not exist
Pre-precalculus Boot Camp: Arithmetic with fractions page http://kunklet.peoplcofedu/ Aug, 0 Arithmetic with fractions To add fractions with the same denominator, add the numerators: () a b + c b = a+c
More information5.2. Adding and Subtracting Polynomials. Objectives. Know the basic definitions for polynomials. Add and subtract polynomials.
Chapter 5 Section 2 5.2 Adding and Subtracting Polynomials Objectives 1 2 Know the basic definitions for polynomials. Add and subtract polynomials. Objective 1 Know the basic definitions for polynomials.
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 informationAlgebra 1 Skills Needed for Success in Math
Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif
More informationTopics Covered in Math 115
Topics Covered in Math 115 Basic Concepts Integer Exponents Use bases and exponents. Evaluate exponential expressions. Apply the product, quotient, and power rules. Polynomial Expressions Perform addition
More information6.2 Multiplying Polynomials
Locker LESSON 6. Multiplying Polynomials PAGE 7 BEGINS HERE Name Class Date 6. Multiplying Polynomials Essential Question: How do you multiply polynomials, and what type of epression is the result? Common
More informationPOLYNOMIALS CHAPTER 2. (A) Main Concepts and Results
CHAPTER POLYNOMIALS (A) Main Concepts and Results Meaning of a Polynomial Degree of a polynomial Coefficients Monomials, Binomials etc. Constant, Linear, Quadratic Polynomials etc. Value of a polynomial
More informationSection 10-1: Laws of Exponents
Section -: Laws of Eponents Learning Outcome Multiply: - ( ) = - - = = To multiply like bases, add eponents, and use common base. Rewrite answer with positive eponent. Learning Outcome Write the reciprocals
More informationAlgebra II Chapter 5: Polynomials and Polynomial Functions Part 1
Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions
More informationReview: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a
Algebra II Notes Unit Si: Polynomials Syllabus Objectives: 6. The student will simplify polynomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a
More information9.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED LESSON
CONDENSED LESSON 9.1 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 solve
More informationMATH98 Intermediate Algebra Practice Test Form A
MATH98 Intermediate Algebra Practice Test Form A MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation. 1) (y - 4) - (y + ) = 3y 1) A)
More informationPolynomials and Polynomial Functions
Unit 5: Polynomials and Polynomial Functions Evaluating Polynomial Functions Objectives: SWBAT identify polynomial functions SWBAT evaluate polynomial functions. SWBAT find the end behaviors of polynomial
More informationA summary of factoring methods
Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 A summary of factoring methods What you need to know already: Basic algebra notation and facts. What you can learn here: What
More information5.1 Monomials. Algebra 2
. Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific
More information