Find two positive factors of 24 whose sum is 10. Make an organized list.

Size: px
Start display at page:

Download "Find two positive factors of 24 whose sum is 10. Make an organized list."

Transcription

1 9.5 Study Guide For use with pages GOAL Factor trinomials of the form x 2 1 bx 1 c. EXAMPLE 1 Factor when b and c are positive Factor x x Find two positive factors of 24 whose sum is 10. Make an organized list. 9.5 of 24 Sum of factors 24, , , correct sum 2, The factors 6 and 4 have a sum of 10, so they are the correct values of p and q. x x (x 1 6)(x 1 4) EXAMPLE 2 CHECK (x 1 6)(x 1 4) 5 x 2 1 4x 1 6x 1 24 Multiply binomials. 5 x x 1 24 Simplify. Factor when b is negative and c is positive Factor w w 1 9. Because b is negative and c is positive, p and q must be negative. of 9 Sum of factors 29, (21) 5210 correct sum 23, (23) 526 The factors 29 and 21 have a sum of 210, so they are the correct values of p and q. w w (x 2 9)(x 2 1) Exercises for Examples 1 and 2 1. x x y 2 1 6y z 2 2 7z

2 9.5 Study Guide continued For use with pages EXAMPLE 3 Factor when b is positive and c is negative Factor k 2 1 6x 2 7. Because c is negative, p and q must have different signs. of 7 Sum of factors 27, , (21) 5 6 correct sum The factors 7 and 21 have a sum of 6, so they are the correct values of p and q. k 2 1 6k (x 1 7)(x 2 1) Exercises for Example 3 4. x x y 2 1 2y z 2 2 5z 2 36 EXAMPLE 4 Solve a polynomial equation Solve the equation h 2 2 4h h 2 2 4h 5 21 Write original equation. h 2 2 4h Subtract 21 from each side. (h 1 3)(h 2 7) 5 0 Factor left side. h or h Zero-product property h523 or h 5 7 Solve for h. The roots of the equation are 23 and 7. Exercise for Example 4 7. Solve the equation x x. 57

3 9.6 Study Guide For use with pages GOAL Factor trinomials of the form ax 2 1 bx 1 c. EXAMPLE 1 Factor when b is negative and c is positive Factor 5n n 1 7. Because b is negative and c is positive, both factors of c must be negative. Make a table to organize your work. You must consider the order of the factors of 7, because the x-terms of the possible factorization are different. of 5 of 7 Possible factorization Middle term when multiplied 1, 5 21, 27 (n 2 1)(5n 2 7) 25n 2 7n 5212n 1, 5 27, 21 (n 2 7)(5n 2 1) 2n 2 35n 5236n 5n n (n 2 1)(5n 2 7) correct EXAMPLE 2 Factor when b is negative and c is negative Factor 3m 2 2 5m Because b is negative and c is negative, p and q must have different signs. of 3 of 22 Possible factorization Middle term when multiplied 1, 3 1, 222 (m 1 1)(3m 2 22) 3m 2 22m 5219m 1, 3 21, 22 (m 2 1)(3m 1 22) 22m 2 3m 5 19m 1, 3 2, 211 (m 1 2)(3m 2 11) 211m 1 6m 525m 1, 3 211, 2 (m 2 11)(3m 1 2) 2m 2 33m 5231m 3m 2 2 5m (m 1 2)(3m 2 11) Exercises for Examples 1 and a a b 2 2 8b c 2 1 5c 2 14 correct

4 9.6 Study Guide continued For use with pages EXAMPLE 3 Factor when a is negative Factor 22x 2 1 9x 2 9. STEP 1 Factor 21 from each term of the trinomial. 22x 2 1 9x (2x 2 2 9x 1 9) STEP 2 Factor the trinomial 2x 2 2 9x 1 9. Because b is negative and c is positive, both factors of c must be negative. Use a table to organize information about the factors of a and c. of 2 of 9 Possible factorization Middle term when multiplied 1, 2 21, 29 (x 2 1)(2x 2 9) 29x 2 2x 5211x 1, 2 29, 21 (x 2 9)(2x 2 1) 2x 2 18x 5219x 1, 2 23, 23 (x 2 3)(2x 2 3) 23x 2 6x 529x correct 22x 2 1 9x (x 2 3)(2x 2 3) 9.6 Exercises for Example r 2 2 7r s 2 1 8s t 2 1 6t

5 9.7 Study Guide For use with pages GOAL Factor special products. Vocabulary The for finding the square of a binomial gives you the for factoring trinomials of the form a 2 1 2ab 1 b 2 and a 2 2 2ab 1 b 2. These are called perfect square trinomials. EXAMPLE 1 Factor the difference of squares a. r r Write as a 2 2 b 2. 5 (r 2 9)(r 1 9) Difference of two squares b. 9s 2 2 4t 2 5 (3s) 2 2 (2t) 2 Write as a 2 2 b 2. 5 (3s 2 2t)(3s 1 2t) Difference of two squares c q 2 5 5( q 2 ) Factor out common factor. 5 5[4 2 2 (5q) 2 ] Write q 2 as a 2 2 b (2 2 5q)(2 1 5q) Difference of two squares Exercises for Example 1 1. m n y z

6 9.7 Study Guide continued For use with pages EXAMPLE 2 Factor perfect square trinomials a. x x x 2 1 2(x)(7) Write as a 2 1 2ab 1 b 2. 5 (x 1 7) 2 Perfect square trinomial b. 144y y (12y) 2 1 2(12y)(5) Write as a 2 1 2ab 1 b 2. 5 (12y 1 5) 2 Perfect square trinomial c. 150z z (25z z 1 1) Factor out common factor. 5 6[(5z) 2 2 2(5z)(1) ] Write 25z z 1 1 as a 2 1 2ab 1 b (5z 2 1) 2 Perfect square trinomial Exercises for Example 2 4. m } 2 m 1 1 } r rs 1 25s x x EXAMPLE 3 Solve a polynomial equation Solve the equation q q Write original equation. q Write left side as a 2 2 b 2. (q 1 10)(q 2 10) 5 0 Difference of two squares q or q Zero-product property q 5210 or q 5 10 Solve for q. The roots of the equation are 210 and 10. Exercises for Example 3 Solve the equation. 7. r r m

Solving Quadratic Equations

Solving Quadratic Equations Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic

More information

Quadratic Formula: - another method for solving quadratic equations (ax 2 + bx + c = 0)

Quadratic Formula: - another method for solving quadratic equations (ax 2 + bx + c = 0) In the previous lesson we showed how to solve quadratic equations that were not factorable and were not perfect squares by making perfect square trinomials using a process called completing the square.

More information

Factoring Trinomials of the Form ax 2 + bx + c, a 1

Factoring 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 information

Polynomial Form. Factored Form. Perfect Squares

Polynomial Form. Factored Form. Perfect Squares We ve seen how to solve quadratic equations (ax 2 + bx + c = 0) by factoring and by extracting square roots, but what if neither of those methods are an option? What do we do with a quadratic equation

More information

5.3. Polynomials and Polynomial Functions

5.3. Polynomials and Polynomial Functions 5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a

More information

UNIT 2 FACTORING. M2 Ch 11 all

UNIT 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 information

MathB65 Ch 4 VII, VIII, IX.notebook. November 06, 2017

MathB65 Ch 4 VII, VIII, IX.notebook. November 06, 2017 Chapter 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest

More information

A-2. Polynomials and Factoring. Section A-2 1

A-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 information

Sections 7.2, 7.3, 4.1

Sections 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 information

Chapter 5: Exponents and Polynomials

Chapter 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 information

Solving Linear Equations

Solving Linear Equations Solving Linear Equations Golden Rule of Algebra: Do unto one side of the equal sign as you will do to the other Whatever you do on one side of the equal sign, you MUST do the same exact thing on the other

More information

Polynomial Form. Factored Form. Perfect Squares

Polynomial Form. Factored Form. Perfect Squares We ve seen how to solve quadratic equations (ax 2 + bx + c = 0) by factoring and by extracting square roots, but what if neither of those methods are an option? What do we do with a quadratic equation

More information

Understand the vocabulary used to describe polynomials Add polynomials Subtract polynomials Graph equations defined by polynomials of degree 2

Understand the vocabulary used to describe polynomials Add polynomials Subtract polynomials Graph equations defined by polynomials of degree 2 Section 5.1: ADDING AND SUBTRACTING POLYNOMIALS When you are done with your homework you should be able to Understand the vocabulary used to describe polynomials Add polynomials Subtract polynomials Graph

More information

Section 7.1 Quadratic Equations

Section 7.1 Quadratic Equations Section 7.1 Quadratic Equations INTRODUCTION In Chapter 2 you learned about solving linear equations. In each of those, the highest power of any variable was 1. We will now take a look at solving quadratic

More information

Section 6.5 A General Factoring Strategy

Section 6.5 A General Factoring Strategy Difference of Two Squares: a 2 b 2 = (a + b)(a b) NOTE: Sum of Two Squares, a 2 b 2, is not factorable Sum and Differences of Two Cubes: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 b 3 = (a b)(a 2 + ab + b

More information

Algebra. Practice Pack

Algebra. Practice Pack Algebra Practice Pack WALCH PUBLISHING Table of Contents Unit 1: Algebra Basics Practice 1 What Are Negative and Positive Numbers?... 1 Practice 2 Larger and Smaller Numbers................ 2 Practice

More information

Maintaining Mathematical Proficiency

Maintaining 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 information

Multiplication of Polynomials

Multiplication 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 information

Review Unit Multiple Choice Identify the choice that best completes the statement or answers the question.

Review 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 information

Lesson 2: Introduction to Variables

Lesson 2: Introduction to Variables Lesson 2: Introduction to Variables Topics and Objectives: Evaluating Algebraic Expressions Some Vocabulary o Variable o Term o Coefficient o Constant o Factor Like Terms o Identifying Like Terms o Combining

More information

Factors of Polynomials Factoring For Experts

Factors of Polynomials Factoring For Experts Factors of Polynomials SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Discussion Group, Note-taking When you factor a polynomial, you rewrite the original polynomial as a product

More information

Controlling the Population

Controlling 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 information

Math 0320 Final Exam Review

Math 0320 Final Exam Review Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:

More information

Solving Equations by Factoring. Solve the quadratic equation x 2 16 by factoring. We write the equation in standard form: x

Solving Equations by Factoring. Solve the quadratic equation x 2 16 by factoring. We write the equation in standard form: x 11.1 E x a m p l e 1 714SECTION 11.1 OBJECTIVES 1. Solve quadratic equations by using the square root method 2. Solve quadratic equations by completing the square Here, we factor the quadratic member of

More information

To solve a radical equation, you must take both sides of an equation to a power.

To solve a radical equation, you must take both sides of an equation to a power. Topic 5 1 Radical Equations A radical equation is an equation with at least one radical expression. There are four types we will cover: x 35 3 4x x 1x 7 3 3 3 x 5 x 1 To solve a radical equation, you must

More information

Order of Operations Practice: 1) =

Order of Operations Practice: 1) = Order of Operations Practice: 1) 24-12 3 + 6 = a) 6 b) 42 c) -6 d) 192 2) 36 + 3 3 (1/9) - 8 (12) = a) 130 b) 171 c) 183 d) 4,764 1 3) Evaluate: 12 2-4 2 ( - ½ ) + 2 (-3) 2 = 4) Evaluate 3y 2 + 8x =, when

More information

Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.2)

Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.2) Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.) Determine if the following functions are polynomials. If so, identify the degree, leading coefficient, and type of polynomial 5 3 1. f ( x) =

More information

Algebra I Unit Report Summary

Algebra 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 information

Math Lecture 18 Notes

Math Lecture 18 Notes Math 1010 - Lecture 18 Notes Dylan Zwick Fall 2009 In our last lecture we talked about how we can add, subtract, and multiply polynomials, and we figured out that, basically, if you can add, subtract,

More information

1.3 Algebraic Expressions. Copyright Cengage Learning. All rights reserved.

1.3 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 information

When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut.

When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut. Squaring a Binomial When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut. Solve. (x 3) 2 Step 1 Square the first term. Rules

More information

Chapter Six. Polynomials. Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring

Chapter Six. Polynomials. Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring Chapter Six Polynomials Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring Properties of Exponents The properties below form the basis

More information

AFM Review Test Review

AFM Review Test Review Name: Class: Date: AFM Review Test Review Multiple Choice Identify the choice that best completes the statement or answers the question. What are the solutions of the inequality?. q + (q ) > 0 q < 3 q

More information

Algebraic Expressions

Algebraic Expressions Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly

More information

review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17

review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17 1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient

More information

evaluate functions, expressed in function notation, given one or more elements in their domains

evaluate functions, expressed in function notation, given one or more elements in their domains Describing Linear Functions A.3 Linear functions, equations, and inequalities. The student writes and represents linear functions in multiple ways, with and without technology. The student demonstrates

More information

Something that can have different values at different times. A variable is usually represented by a letter in algebraic expressions.

Something that can have different values at different times. A variable is usually represented by a letter in algebraic expressions. Lesson Objectives: Students will be able to define, recognize and use the following terms in the context of polynomials: o Constant o Variable o Monomial o Binomial o Trinomial o Polynomial o Numerical

More information

Introduction. Adding and Subtracting Polynomials

Introduction. Adding and Subtracting Polynomials Introduction Polynomials can be added and subtracted like real numbers. Adding and subtracting polynomials is a way to simplify expressions. It can also allow us to find a shorter way to represent a sum

More information

UNIT 5 QUADRATIC FUNCTIONS Lesson 1: Interpreting Structure in Expressions Instruction

UNIT 5 QUADRATIC FUNCTIONS Lesson 1: Interpreting Structure in Expressions Instruction Prerequisite Skills This lesson requires the use of the following skills: evaluating expressions using the order of operations evaluating expressions for a given value identifying parts of an expression

More information

MathB65 Ch 4 IV, V, VI.notebook. October 31, 2017

MathB65 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 information

Math 1 Variable Manipulation Part 6 Polynomials

Math 1 Variable Manipulation Part 6 Polynomials Name: Math 1 Variable Manipulation Part 6 Polynomials Date: 1 VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does not have

More information

Topic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3

Topic 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 information

Adding and Subtracting Polynomials Add and Subtract Polynomials by doing the following: Combine like terms

Adding and Subtracting Polynomials Add and Subtract Polynomials by doing the following: Combine like terms POLYNOMIALS AND POLYNOMIAL OPERATIONS STUDY GUIDE Polynomials Polynomials are classified by two different categories: by the number of terms, and the degree of the leading exponent. Number Classification

More information

5.1, 5.2, 5.3 Properites of Exponents last revised 6/7/2014. c = Properites of Exponents. *Simplify each of the following:

5.1, 5.2, 5.3 Properites of Exponents last revised 6/7/2014. c = Properites of Exponents. *Simplify each of the following: 48 5.1, 5.2, 5.3 Properites of Exponents last revised 6/7/2014 Properites of Exponents 1. x a x b = x a+b *Simplify each of the following: a. x 4 x 8 = b. x 5 x 7 x = 2. xa xb = xa b c. 5 6 5 11 = d. x14

More information

Factoring x 2 + bx + c

Factoring x 2 + bx + c 7.5 Factoring x 2 + bx + c Essential Question How can you use algebra tiles to factor the trinomial x 2 + bx + c into the product of two binomials? Finding Binomial Factors Work with a partner. Use algebra

More information

Unit 5 AB Quadratic Expressions and Equations 1/9/2017 2/8/2017

Unit 5 AB Quadratic Expressions and Equations 1/9/2017 2/8/2017 Unit 5 AB Quadratic Expressions and Equations 1/9/2017 2/8/2017 Name: By the end of this unit, you will be able to Add, subtract, and multiply polynomials Solve equations involving the products of monomials

More information

Lesson 6. Diana Pell. Monday, March 17. Section 4.1: Solve Linear Inequalities Using Properties of Inequality

Lesson 6. Diana Pell. Monday, March 17. Section 4.1: Solve Linear Inequalities Using Properties of Inequality Lesson 6 Diana Pell Monday, March 17 Section 4.1: Solve Linear Inequalities Using Properties of Inequality Example 1. Solve each inequality. Graph the solution set and write it using interval notation.

More information

QUADRATIC FUNCTIONS AND MODELS

QUADRATIC FUNCTIONS AND MODELS QUADRATIC FUNCTIONS AND MODELS What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results to sketch graphs of functions. Find minimum and

More information

UNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction

UNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction Prerequisite Skills This lesson requires the use of the following skills: simplifying radicals working with complex numbers Introduction You can determine how far a ladder will extend from the base of

More information

Beginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions

Beginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:

More information

Lecture 27. Quadratic Formula

Lecture 27. Quadratic Formula Lecture 7 Quadratic Formula Goal: to solve any quadratic equation Quadratic formula 3 Plan Completing the square 5 Completing the square 6 Proving quadratic formula 7 Proving quadratic formula 8 Proving

More information

Vocabulary. Term Page Definition Clarifying Example. binomial. cubic. degree of a monomial. degree of a polynomial

Vocabulary. Term Page Definition Clarifying Example. binomial. cubic. degree of a monomial. degree of a polynomial CHAPTER 7 Vocabulary This table contains important vocabulary terms from Chapter 7. As you work through the chapter, fill in the page number, definition, and a clarifying example for each term. binomial

More information

Algebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials

Algebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials Algebra 1: Hutschenreuter Chapter 10 Notes Name 10.1 Adding and Subtracting Polynomials Polynomial- an expression where terms are being either added and/or subtracted together Ex: 6x 4 + 3x 3 + 5x 2 +

More information

( ) Chapter 6 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3.

( ) Chapter 6 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3. Chapter 6 Exercise Set 6.1 1. A prime number is an integer greater than 1 that has exactly two factors, itself and 1. 3. To factor an expression means to write the expression as the product of factors.

More information

Lesson 3: Polynomials and Exponents, Part 1

Lesson 3: Polynomials and Exponents, Part 1 Lesson 2: Introduction to Variables Assessment Lesson 3: Polynomials and Exponents, Part 1 When working with algebraic expressions, variables raised to a power play a major role. In this lesson, we look

More information

Chapter 1.6. Perform Operations with Complex Numbers

Chapter 1.6. Perform Operations with Complex Numbers Chapter 1.6 Perform Operations with Complex Numbers EXAMPLE Warm-Up 1 Exercises Solve a quadratic equation Solve 2x 2 + 11 = 37. 2x 2 + 11 = 37 2x 2 = 48 Write original equation. Subtract 11 from each

More information

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!! 1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a

More information

9-8 Completing the Square

9-8 Completing the Square In the previous lesson, you solved quadratic equations by isolating x 2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square. When

More information

I CAN classify polynomials by degree and by the number of terms.

I 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 information

Warm-Up. Use long division to divide 5 into

Warm-Up. Use long division to divide 5 into Warm-Up Use long division to divide 5 into 3462. 692 5 3462-30 46-45 12-10 2 Warm-Up Use long division to divide 5 into 3462. Divisor 692 5 3462-30 46-45 12-10 2 Quotient Dividend Remainder Warm-Up Use

More information

B. Complex number have a Real part and an Imaginary part. 1. written as a + bi some Examples: 2+3i; 7+0i; 0+5i

B. Complex number have a Real part and an Imaginary part. 1. written as a + bi some Examples: 2+3i; 7+0i; 0+5i Section 11.8 Complex Numbers I. The Complex Number system A. The number i = -1 1. 9 and 24 B. Complex number have a Real part and an Imaginary part II. Powers of i 1. written as a + bi some Examples: 2+3i;

More information

Section 1.7: Solving Equations by Factoring

Section 1.7: Solving Equations by Factoring Objective: Solve equations by factoring and using the zero product rule. When solving linear equations such as x 5 1, we can solve for the variable directly by adding 5 and dividing by to get 1. However,

More information

P.5 Solving Equations

P.5 Solving Equations PRC Ch P_5.notebook P.5 Solving Equations What you should learn How to solve linear equations How to solve quadratic equations equations How to solve polynomial equations of degree three or higher How

More information

THE RING OF POLYNOMIALS. Special Products and Factoring

THE RING OF POLYNOMIALS. Special Products and Factoring THE RING OF POLYNOMIALS Special Products and Factoring Special Products and Factoring Upon completion, you should be able to Find special products Factor a polynomial completely Special Products - rules

More information

HONORS GEOMETRY Summer Skills Set

HONORS GEOMETRY Summer Skills Set HONORS GEOMETRY Summer Skills Set Algebra Concepts Adding and Subtracting Rational Numbers To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference

More information

RAVEN S MANITOBA GRADE 10 INTRODUCTION TO APPLIED AND PRE CALCULUS MATHEMATICS (20S)

RAVEN S MANITOBA GRADE 10 INTRODUCTION TO APPLIED AND PRE CALCULUS MATHEMATICS (20S) RAVEN S MANITOBA GRADE 10 INTRODUCTION TO APPLIED AND PRE CALCULUS MATHEMATICS (20S) LINKED DIRECTLY TO NEW CURRICULUM REQUIREMENTS FROM THE WESTERN PROTOCOLS FOR 2008 AND BEYOND STUDENT GUIDE AND RESOURCE

More information

8-1: Adding and Subtracting Polynomials

8-1: Adding and Subtracting Polynomials 8-1: Adding and Subtracting Polynomials Objective: To classify, add, and subtract polynomials Warm Up: Simplify each expression. 1. x 3 7 x 9. 6(3x 4) 3. 7 ( x 8) 4 4. 5(4x (8x 6) monomial - A real number,

More information

For Your Notebook E XAMPLE 1. Factor when b and c are positive KEY CONCEPT. CHECK (x 1 9)(x 1 2) 5 x 2 1 2x 1 9x Factoring x 2 1 bx 1 c

For Your Notebook E XAMPLE 1. Factor when b and c are positive KEY CONCEPT. CHECK (x 1 9)(x 1 2) 5 x 2 1 2x 1 9x Factoring x 2 1 bx 1 c 9.5 Factor x2 1 bx 1 c Before You factored out the greatest common monomial factor. Now You will factor trinomials of the form x 2 1 bx 1 c. Why So you can find the dimensions of figures, as in Ex. 61.

More information

Lesson 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 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 information

PERT Practice Test #2

PERT Practice Test #2 Class: Date: PERT Practice Test #2 Multiple Choice Identify the choice that best completes the statement or answers the question. Ê 1. What is the quotient of 6y 6 9y 4 + 12y 2 ˆ Ê 3y 2 ˆ? a. 2y 4 + 3y

More information

correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1

correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 McDougal Littell Algebra 1 2007 correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 The main goal of Algebra is to

More information

Mathematics Diagnostic Examination Guidance

Mathematics Diagnostic Examination Guidance Mathematics Diagnostic Examination Guidance Examination Overview The mathematics examination will be 45 minutes long and will be worth 50 points. There will be three sections on the examination: Section

More information

5.2. Adding and Subtracting Polynomials. Objectives. Know the basic definitions for polynomials. Add and subtract polynomials.

5.2. 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

Adding and Subtracting Polynomials

Adding 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 information

Prerequisites. Copyright Cengage Learning. All rights reserved.

Prerequisites. 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 information

MULTIPLYING TRINOMIALS

MULTIPLYING TRINOMIALS Name: Date: 1 Math 2 Variable Manipulation Part 4 Polynomials B MULTIPLYING TRINOMIALS Multiplying trinomials is the same process as multiplying binomials except for there are more terms to multiply than

More information

Unit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power.

Unit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power. MODULE 1 1 Polynomial A function that contains 1 or more or terms. The variables may be to any non-negative power. 1 Modeling Mathematical modeling is the process of using, and to represent real world

More information

Chapter 9 Quadratic Functions and Equations

Chapter 9 Quadratic Functions and Equations Chapter 9 Quadratic Functions and Equations 1 9 1Quadratic Graphs and their properties U shaped graph such as the one at the right is called a parabola. A parabola can open upward or downward. A parabola

More information

Math 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 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 information

Teacher's Page. Mar 20 8:26 AM

Teacher's Page. Mar 20 8:26 AM Teacher's Page Unit 4.1 in two parts: Part 1: Polynomials and Part 2: Quadratics Benchmarks: A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity

More information

Unit 2: Polynomials Guided Notes

Unit 2: Polynomials Guided Notes Unit 2: Polynomials Guided Notes Name Period **If found, please return to Mrs. Brandley s room, M 8.** Self Assessment The following are the concepts you should know by the end of Unit 1. Periodically

More information

MAFS Algebra 1. Polynomials. Day 15 - Student Packet

MAFS Algebra 1. Polynomials. Day 15 - Student Packet MAFS Algebra 1 Polynomials Day 15 - Student Packet Day 15: Polynomials MAFS.91.A-SSE.1., MAFS.91.A-SSE..3a,b, MAFS.91.A-APR..3, MAFS.91.F-IF.3.7c I CAN rewrite algebraic expressions in different equivalent

More information

27 Wyner Math 2 Spring 2019

27 Wyner Math 2 Spring 2019 27 Wyner Math 2 Spring 2019 CHAPTER SIX: POLYNOMIALS Review January 25 Test February 8 Thorough understanding and fluency of the concepts and methods in this chapter is a cornerstone to success in the

More information

EXAMPLE 1. a. Add 2x 3 5x 2 + 3x 9 and x 3 + 6x in a vertical format. SOLUTION. a. 2x 3 5x 2 + 3x 9 + x 3 + 6x x 3 + x 2 + 3x + 2

EXAMPLE 1. a. Add 2x 3 5x 2 + 3x 9 and x 3 + 6x in a vertical format. SOLUTION. a. 2x 3 5x 2 + 3x 9 + x 3 + 6x x 3 + x 2 + 3x + 2 EXAMPLE 1 Add polynomials vertically and horizontally a. Add 2x 3 5x 2 + 3x 9 and x 3 + 6x 2 + 11 in a vertical format. a. 2x 3 5x 2 + 3x 9 + x 3 + 6x 2 + 11 3x 3 + x 2 + 3x + 2 EXAMPLE 1 Add polynomials

More information

Lecture Guide. Math 90 - Intermediate Algebra. Stephen Toner. Intermediate Algebra, 3rd edition. Miller, O'Neill, & Hyde. Victor Valley College

Lecture Guide. Math 90 - Intermediate Algebra. Stephen Toner. Intermediate Algebra, 3rd edition. Miller, O'Neill, & Hyde. Victor Valley College Lecture Guide Math 90 - Intermediate Algebra to accompany Intermediate Algebra, 3rd edition Miller, O'Neill, & Hyde Prepared by Stephen Toner Victor Valley College Last updated: 4/17/16 5.1 Exponents &

More information

{ independent variable some property or restriction about independent variable } where the vertical line is read such that.

{ 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 information

Algebra 1 Seamless Curriculum Guide

Algebra 1 Seamless Curriculum Guide QUALITY STANDARD #1: REAL NUMBERS AND THEIR PROPERTIES 1.1 The student will understand the properties of real numbers. o Identify the subsets of real numbers o Addition- commutative, associative, identity,

More information

Gaithersburg High School Summer 2018 Math Packet For Rising Algebra 2/Honors Algebra 2 Students

Gaithersburg High School Summer 2018 Math Packet For Rising Algebra 2/Honors Algebra 2 Students Gaithersburg High School Math Packet For Rising Algebra 2/Honors Algebra 2 Students 1 This packet is an optional review of the skills that will help you be successful in Algebra 2 in the fall. By completing

More information

6.1 Using Properties of Exponents 1. Use properties of exponents to evaluate and simplify expressions involving powers. Product of Powers Property

6.1 Using Properties of Exponents 1. Use properties of exponents to evaluate and simplify expressions involving powers. Product of Powers Property 6.1 Using Properties of Exponents Objectives 1. Use properties of exponents to evaluate and simplify expressions involving powers. 2. Use exponents and scientific notation to solve real life problems.

More information

Math 2 Variable Manipulation Part 3 Polynomials A

Math 2 Variable Manipulation Part 3 Polynomials A Math 2 Variable Manipulation Part 3 Polynomials A 1 MATH 1 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 not

More information

A monomial is measured by its degree To find its degree, we add up the exponents of all the variables of the monomial.

A monomial is measured by its degree To find its degree, we add up the exponents of all the variables of the monomial. UNIT 6 POLYNOMIALS Polynomial (Definition) A monomial or a sum of monomials. A monomial is measured by its degree To find its degree, we add up the exponents of all the variables of the monomial. Ex. 2

More information

Day 7: Polynomials MAFS.912.A-SSE.1.2, MAFS.912.A-SSE.2.3a,b, MAFS.912.A-APR.2.3, MAFS.912.F-IF.3.7c

Day 7: Polynomials MAFS.912.A-SSE.1.2, MAFS.912.A-SSE.2.3a,b, MAFS.912.A-APR.2.3, MAFS.912.F-IF.3.7c Day 7: Polynomials MAFS.91.A-SSE.1., MAFS.91.A-SSE..3a,b, MAFS.91.A-APR..3, MAFS.91.F-IF.3.7c I CAN rewrite algebraic expressions in different equivalent forms using factoring techniques use equivalent

More information

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1

Algebra 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 information

Intensive Math-Algebra I Mini-Lesson MA.912.A.4.3

Intensive 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 information

Unit 5 Quadratic Expressions and Equations

Unit 5 Quadratic Expressions and Equations Unit 5 Quadratic Expressions and Equations Test Date: Name: By the end of this unit, you will be able to Add, subtract, and multiply polynomials Solve equations involving the products of monomials and

More information

Unit 2: Polynomials Guided Notes

Unit 2: Polynomials Guided Notes Unit 2: Polynomials Guided Notes Name Period **If found, please return to Mrs. Brandley s room, M 8.** Self Assessment The following are the concepts you should know by the end of Unit 1. Periodically

More information

To factor an expression means to write it as a product of factors instead of a sum of terms. The expression 3x

To factor an expression means to write it as a product of factors instead of a sum of terms. The expression 3x Factoring trinomials In general, we are factoring ax + bx + c where a, b, and c are real numbers. To factor an expression means to write it as a product of factors instead of a sum of terms. The expression

More information

Day 28 linear functions Day 29 linear functions. integers Day 30 non-linear functions Day 31 non-linear functions. Multiply and divide integers Day

Day 28 linear functions Day 29 linear functions. integers Day 30 non-linear functions Day 31 non-linear functions. Multiply and divide integers Day Algebra 1 Credits:1 Prerequisite:Pre-algebra Recommended:8th, 9th Course Description:Students will engage in real world and hands-on problem solving while using their developing skills in algebra. Students

More information

Unit 13: Polynomials and Exponents

Unit 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 information