Chapter 5 - Polynomials and Polynomial Functions

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 5 - Polynomials and Polynomial Functions"

Transcription

1 Math Spring 2009 Chapter 5 - Polynomials and Polynomial Functions 5.1 Addition and Subtraction of Polynomials Definition 1. A polynomial is a finite sum of terms in which all variables have whole number exponents (recall whole numbers are positive and include 0) and no variable appears in the denominator. EX The following are not polynomials. State why not: x 1 2, 1 x + 1, x 2 2. Here are a couple examples of polynomials: (a) 7x 6 + 5x 4 2x (b) x 3 y 2 2x 2 y + 9xyz 5 Recall that the degree of a term is the sum of the exponents of the variables. The leading term of a polynomial is the term of highest degree and the leading coefficient is the coefficient of the leading term. EX 2. For the polynomial 7x 6 + 5x 4 2x we have: Degree of the polynomial: 6 Leading Term: 7x 6 Leading Coefficient: -7 Classifying Polynomials We can classify polynomials in a couple of ways. First by the number of terms: Terminology Number of terms Examples Monomial one term 4x 2, 7x 3 y 2, 6 Binomial two terms x 1, x 3 y Trinomial three terms x 2 + 2x 1, x 3 y + 2y x We can also categorize polynomials by the degree of the polynomials Linear polynomial - polynomial of degree 0 or 1. Quadratic Polynomial - polynomial of degree 2. Cubic Polynomial - polynomial of degree 3. After degree three we usually just refer to the polynomial as n-th degree, ie 4th degree, 5th degree, etc. We generally will write polnomials in descending order in terms of degree. EX 3. For example we would write x + 6x 2 + 2x 3 1 as 2x 3 + 6x 2 + x 1 Where we start with the highest degree term and continue in descending order. 1

2 5.1.1 Polynomial Functions Polynomials can also be written as function (using function notation). EX 4. The following are polynomial functions. f(x) = x 2 + 3x + 2 P (x) = 2x 3 3x 2 + x 7 When written as a function, we can graph polynomials. EX Graph y = x 2 The graph we get is called a parabola. All quadratic polynomials have parabolas as graphs. In this case the parabola opens upwards. Whenever the leading coefficient is positive the parabola opens upwards. 2. Graph y = x 2 We notice that we now have a negative leading coefficient and the parabola opens downwards Add and Subtract Polynomials To add or subtract polynomials we just combine like terms EX 6. Simplify the following: 1. Add (3x 2 2x + 7) + (2x 2 + x 3) 2. Add (2x 2 y 4xy + 7) + (3x 2 y + 5xy 2y 8) 2

3 3. Subtract ( x 2 + 5x 13) from (x 3 2x + 9) 4. Simplify 8x 2 y 7xy (x 2 y 2xy 2 + 6y) 5.2 Multiplication of Polynomials We learn how to multiply polynomials in steps Multiply a Polynomial by a Monomial To begine we recall product rule for exponents a m a n = a m+n EX 7. Monomial Monomial: Multiply (5x 2 y)(6x 8 y 2 ) In order to do more complicated products we need to recall the distributive property a(b + c + d + + n) = a b + a b + a c + a d + + a n EX 8. Multiply: 3x 2 y(2xy + 7xy 2 8) Multiply a Binomial by a Binomial Abstractly we multiply binomial times a binomial as follows: (a + b)(c + d) = (a + b) c + (a + b) d = a c + b c + a d + b d However we will generally use the FOIL method. FOIL stands for: some examples EX 9. Multiply: 1. (4x 3)(x + 2) F - First O - Outer I - Inner L - Last Let s look at 3

4 2. (3x 2 + 4)(2x 5) Multiply a Polynomial by a Polynomial Let s look at a couple examples. EX 10. Multiply: 1. Multiply x x by 4x Multiply 2x 2 + 5xy 4y 2 by x + 3y Some Formulas and Special Cases The following situations come up often enough that we look at them seperately: (a + b) 2 = (a + b)(a + b) = a 2 + 2ab + b 2 (a b) 2 = (a b)(a b) = a 2 2ab + b 2 and (a + b)(a b) = a 2 b 2 EX 11. Multiply: 1. (x + 2) 2 (warning: (x + 2) 2 x 2 + 4) 2. (3x 2 7y) 2 3. [x + (y 3)] 2 4

5 4. (4x )(4x 2 5 ) 5. (7x + y 5 )(7x y 5 ) 5.3 Division of Polynomials Division by Monomials Recall the following properties that we will use: and the quotient rule for exponents EX 12. Divide: 1. 12x 5 y 8 3x 2 y 6 a + b c a m a n = a c + b c = am n 2. 11r 7 s 1 0 4rs x 2 15x+8 3x Divide a Polynomial by a Binomial To divide two polynomials we use a procedure very similar to long division. Let s look at examples: EX 13. Divide: 1. x 2 +11x+28 x x 2 13x+2 2x 5 5

6 3. 6x 2 16x+4x x Factoring Recall, in section 5.2 we learned how to multiply two polynomials together. In this section we will learn how to undo that multiplication. When we do this, we will say we are factoring a polynomial. Factoring is the opposite of multiplying Find the Greatest Common Factor We will first learn to factor a monomial (single term) from a polynomial. The greatest common factor (GCF) is the product of the factors common to all terms in the polynomial. EX Consider the terms 4x 4 y 2 z, 6x 3 y 4, 9xz 8. Find the greatest common factor (GCF) among the terms. 2. Find the GCF: 28(x 4) 2, 7(x 4) 5, 14(x 4) 3 Now that we know how to find the GCF, let s factor a monomial from a polynomial: Steps to factoring out the GCF 1. Determine the GCF of all terms in the polynomial. 2. Write each term as the product of the GCF and another factor. 3. Use the distributive property to factor out the GCF. EX Factor 6x 5 4x x 3 6

7 2. Factor 33x 4 y + 15x 3 y 2 6x 2 y 3 3. Factor 3b 3 + 6b 2 30b 4. Factor 9x(2x 3) + 7(2x 3) 5. Factor 15(3x + 7) + 5(3x + 7) 6. Factor (5x + 1)(m + n) (2x 1)(m + n) Factor by Grouping When a polynomial contains four terms, it may be possible to factor by grouping. Let s see how: EX Factor mx + my + nx + ny 2. Factor x 3 + 6x 2 4x 24 7

8 3. Factor x 3 4x+6x 2 24 (notice this is the same as above but with the middle terms switched) 5.5 Factoring Trinomials Recall that a trinomial is a polynomial with three terms. We will specifically factor polynomials of the form: ax 2 + bx + c We will look at a couple methods for doing this. First, if a = 1 we can use a guess and check method: EX Factor x 2 x Factor m 2 15m + 56 A more systematic method is to factor trinomials by grouping. Let s look at the steps for this: If our polynomial is of the form ax 2 + bx + c To factor by grouping: 1. Find two numbers whose product is a c and whose sum is b. 2. Rewrite the middle term, bx, using the numbers found in step Factor by grouping. EX 18. Factor p 2 3pq 18q 2 WARNING: The fisrt step when factoring any trinomial is to determine whether all three terms have a common factor. If they do, you must first factor out the common factor. EX

9 1. Factor 2x 5 8x 4 42x 3 2. Factor 4x 2 13x Factor 10x 2 7x Factoring Trinomials Using Substitution Sometimes we can use our methods developed above to factor more complicated polynomials. EX Factor y 4 + y Factor 5z 6 13z Factor 2(x + 5) 2 5(x + 5) 12 9

10 5.6 Special Formulas We look at several important and common forms of polynomials and formulas to factor them Difference of Squares The Formula: a 2 b 2 = (a + b)(a b) Check this formula by foiling the right hand side. EX Factor 81m 2 4n 2 2. Factor 3x 4 48y 4 3. Factor (x 7) 2 25 Warning: We can NOT factor polynomials of the form a 2 + b Perfect Square Trinomials The Formulas: a 2 + 2ab + b 2 = (a + b) 2 a 2 2ab + b 2 = (a b) 2 EX Factor x 2 10x

11 2. Factor 16m 4 24m Facotr 9a ab + 25b The Sum and Difference of Two Cubes The Formulas: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 b 3 = (a b)(a 2 + ab + b 2 ) EX Factor x Factor 64a 3 27b 6 3. Factor 5x 3 40y Review of Factoring See chart on page 354 and handout. 5.8 Polynomial Equations A polynomial equation is an equation where two polynomials are set equal to each other. 11

12 EX 24. The following are examples of polynomial equations: x 3 + 3x = 2x 5 5x 2 + 3x 2 = 0 We will primarily focus on quadratic equations, ones with highest degree two. For these we will write them in standard form: ax 2 + bx + c = Zero Factor Property For all real numbers a and b, if a b = 0 then either a = 0 or b = 0, or both a and b are zero. In other words, if the product of two numbers is zero, then one or both of the numbers is zero. Why is this useful? EX 25. Use the zero factor property to solve the equation Use Factoring to Solve Equations Steps to solving quadratic equations: 1. Write the equation in standard form. 2. Combine like terms and factor (x + 6)(x 1) = 0 3. Set each factor (with a variable) equal to zero and solve each equation. 4. Check the solution in the original equation. EX Solve 7x 2 = 56x 2. Solve (5x 1)(x + 2) = 6x 3. Solve 3x 2 + 7x 24 = x 12

13 4. Solve 4r r 2 3r = 0 5. A rock is thrown upward at a speed of 32 ft/sec from the top of a 128-foot cliff. The height, h, of the rock above the ground at any time, t, in seconds, is determined by the function h(t) = 16t t Find the time it takes for the rock to hit the ground after it is thrown. 13

Mth 95 Module 2 Spring 2014

Mth 95 Module 2 Spring 2014 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

More information

1.3 Polynomials and Factoring

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 information

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

More information

6.1 Add & Subtract Polynomial Expression & Functions

6.1 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 information

POLYNOMIALS and FACTORING

POLYNOMIALS and FACTORING POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

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

More information

Unit 3 Polynomials Study Guide

Unit 3 Polynomials Study Guide Unit Polynomials Study Guide 7-5 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 information

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

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

Algebra Unit 6 Syllabus revised 2/27/13 Exponents and Polynomials

Algebra 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. Pre-assessment: Exponents, Fractions, and Polynomial Expressions Lesson: Pages

More information

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

More information

Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).

Definitions 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 information

Section 6.1 Factoring Expressions

Section 6.1 Factoring Expressions Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what

More information

Name Intro to Algebra 2. Unit 1: Polynomials and Factoring

Name 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 information

Factoring and Applications

Factoring 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 information

1.3 Algebraic Expressions

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

More information

Greatest Common Factor (GCF) Factoring

Greatest Common Factor (GCF) Factoring Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication

More information

Tool 1. Greatest Common Factor (GCF)

Tool 1. Greatest Common Factor (GCF) Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When

More information

Factoring Polynomials and Solving Quadratic Equations

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

More information

Operations with Algebraic Expressions: Multiplication of Polynomials

Operations with Algebraic Expressions: Multiplication of Polynomials Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the

More information

MATH 65 NOTEBOOK CERTIFICATIONS

MATH 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 information

Chapter R.4 Factoring Polynomials

Chapter R.4 Factoring Polynomials Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x

More information

MATH 10034 Fundamental Mathematics IV

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

More information

MATH 90 CHAPTER 6 Name:.

MATH 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 information

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2

Unit 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 information

Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF

Polynomials. 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 information

Algebra 2 PreAP. Name Period

Algebra 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 information

Mathematics Placement

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

More information

FACTORING POLYNOMIALS

FACTORING POLYNOMIALS 296 (5-40) 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 information

This 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 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 by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

More information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

Factoring Trinomials: The ac Method

Factoring 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 information

Using the ac Method to Factor

Using 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 trial-and-error

More information

Unit: Polynomials and Factoring

Unit: 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 information

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

Determinants can be used to solve a linear system of equations using Cramer s Rule. 2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

More information

Factoring Flow Chart

Factoring 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 information

( ) FACTORING. x In this polynomial the only variable in common to all is x.

( ) 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 information

Factoring Polynomials

Factoring 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 information

Factoring Polynomials

Factoring 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 information

Sect 6.7 - Solving Equations Using the Zero Product Rule

Sect 6.7 - Solving Equations Using the Zero Product Rule Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred

More information

By reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.

By 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 information

When factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.

When 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 information

Factoring (pp. 1 of 4)

Factoring (pp. 1 of 4) Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common

More information

Polynomials. 4-4 to 4-8

Polynomials. 4-4 to 4-8 Polynomials 4-4 to 4-8 Learning Objectives 4-4 Polynomials Monomials, binomials, and trinomials Degree of a polynomials Evaluating polynomials functions Polynomials Polynomials are sums of these "variables

More information

Algebra Cheat Sheets

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

More information

A Systematic Approach to Factoring

A Systematic Approach to Factoring A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool

More information

Factoring Special Polynomials

Factoring 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 information

Monomials with the same variables to the same powers are called like terms, If monomials are like terms only their coefficients can differ.

Monomials 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 information

Chapter 7 - Roots, Radicals, and Complex Numbers

Chapter 7 - Roots, Radicals, and Complex Numbers Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

More information

Pre-Calculus II Factoring and Operations on Polynomials

Pre-Calculus II Factoring and Operations on Polynomials Factoring... 1 Polynomials...1 Addition of Polynomials... 1 Subtraction of Polynomials...1 Multiplication of Polynomials... Multiplying a monomial by a monomial... Multiplying a monomial by a polynomial...

More information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

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

More information

LESSON 6.2 POLYNOMIAL OPERATIONS I

LESSON 6.2 POLYNOMIAL OPERATIONS I LESSON 6.2 POLYNOMIAL OPERATIONS I Overview In business, people use algebra everyday to find unknown quantities. For example, a manufacturer may use algebra to determine a product s selling price in order

More information

In algebra, factor by rewriting a polynomial as a product of lower-degree polynomials

In algebra, factor by rewriting a polynomial as a product of lower-degree polynomials Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

Factoring Algebra- Chapter 8B Assignment Sheet

Factoring 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 information

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

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

More information

Chapter 4. Polynomials

Chapter 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 information

Math 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. 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 + 12-2(x 2 + 2x 1) = -2x 2 4x + 2 Multiply the following: 1. 5(x

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

6.3 FACTORING ax 2 bx c WITH a 1

6.3 FACTORING ax 2 bx c WITH a 1 290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100

More information

Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.

Lagrange 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 information

7-8 Multiplying Polynomials

7-8 Multiplying Polynomials 7-8 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 information

Polynomial Expression

Polynomial Expression DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

More information

SPECIAL PRODUCTS AND FACTORS

SPECIAL PRODUCTS AND FACTORS CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

More information

Name Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE

Name 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 information

Veterans Upward Bound Algebra I Concepts - Honors

Veterans Upward Bound Algebra I Concepts - Honors Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring

More information

Monomial. 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

Monomial. 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 information

A. Factoring out the Greatest Common Factor.

A. 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 information

Polynomials. Solving Equations by Using the Zero Product Rule

Polynomials. Solving Equations by Using the Zero Product Rule mil23264_ch05_303-396 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 information

Factoring. Factoring Monomials Monomials can often be factored in more than one way.

Factoring. 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 information

Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

More information

Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai

Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 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 information

FACTORING OUT COMMON FACTORS

FACTORING OUT COMMON FACTORS 278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the

More information

Actually, 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

Actually, 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 information

DMA 060 Polynomials and Quadratic Applications

DMA 060 Polynomials and Quadratic Applications DMA 060 Polynomials and Quadratic Applications Brief Description This course provides a conceptual study of problems involving graphic and algebraic representations of quadratics. Topics include basic

More information

Difference of Squares and Perfect Square Trinomials

Difference 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 information

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

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions. Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

More information

6.1 The Greatest Common Factor; Factoring by Grouping

6.1 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 information

AIP Factoring Practice/Help

AIP Factoring Practice/Help The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There

More information

UNIT TWO POLYNOMIALS MATH 421A 22 HOURS. Revised May 2, 00

UNIT TWO POLYNOMIALS MATH 421A 22 HOURS. Revised May 2, 00 UNIT TWO POLYNOMIALS MATH 421A 22 HOURS Revised May 2, 00 38 UNIT 2: POLYNOMIALS Previous Knowledge: With the implementation of APEF Mathematics at the intermediate level, students should be able to: -

More information

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1 5.7 Factoring ax 2 bx c (5-49) 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 information

7-6. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content

7-6. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content 7-6 Choosing a Factoring Model Extension: Factoring Polynomials with More Than One Variable Essential question: How can you factor polynomials with more than one variable? What is the connection between

More information

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

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

More information

Factoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1)

Factoring 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 information

HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

More information

ModuMath Algebra Lessons

ModuMath 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 information

Algebra Tiles Activity 1: Adding Integers

Algebra 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 information

Algebra 1 Chapter 08 review

Algebra 1 Chapter 08 review Name: Class: Date: ID: A Algebra 1 Chapter 08 review Multiple Choice Identify the choice that best completes the statement or answers the question. Simplify the difference. 1. (4w 2 4w 8) (2w 2 + 3w 6)

More information

Sect 6.1 - Greatest Common Factor and Factoring by Grouping

Sect 6.1 - Greatest Common Factor and Factoring by Grouping Sect 6.1 - Greatest Common Factor and Factoring by Grouping Our goal in this chapter is to solve non-linear equations by breaking them down into a series of linear equations that we can solve. To do this,

More information

Developmental Math Course Outcomes and Objectives

Developmental Math Course Outcomes and Objectives Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/Pre-Algebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and

More information

Factoring Quadratic Expressions

Factoring 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 information

Factoring a Difference of Two Squares. Factoring a Difference of Two Squares

Factoring a Difference of Two Squares. Factoring a Difference of Two Squares 284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this

More information

Factor Polynomials Completely

Factor 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 information

BEGINNING ALGEBRA ACKNOWLEDMENTS

BEGINNING ALGEBRA ACKNOWLEDMENTS BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science

More information

Polynomial Equations and Factoring

Polynomial Equations and Factoring 7 Polynomial Equations and Factoring 7.1 Adding and Subtracting Polynomials 7.2 Multiplying Polynomials 7.3 Special Products of Polynomials 7.4 Dividing Polynomials 7.5 Solving Polynomial Equations in

More information

How to get the Simplified Expanded Form of a polynomial, II

How to get the Simplified Expanded Form of a polynomial, II How to get the Simplified Expanded Form of a polynomial, II Nikos Apostolakis September 29, 2010 Recall. Recall that the distributive law states that multiplication distributes over addition and subtraction:

More information

Factoring Trinomials of the Form x 2 bx c

Factoring 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 information

Higher Education Math Placement

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

More information

Simplifying Algebraic Fractions

Simplifying Algebraic Fractions 5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions

More information