Chapter 8 Polynomials and Factoring

Similar documents
Section 9.1: Add and Subtract Polynomials. The number part of a term with a variable part.

The number part of a term with a variable part. Terms that have the same variable parts. Constant terms are also like terms.

7-7 Multiplying Polynomials

5.3. Polynomials and Polynomial Functions

Chapter 5: Exponents and Polynomials

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.

Algebra 1. Math Review Packet. Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals

Solving Multi-Step Equations

POLYNOMIAL EXPRESSIONS PART 1

Looking Ahead to Chapter 10

Additional Exercises 7.1 Form I The Greatest Common Factor and Factoring by Grouping

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

Fair Game Review. Chapter 7. Simplify the expression. Write an expression for the perimeter of the figure

Polynomials 370 UNIT 10 WORKING WITH POLYNOMIALS. The railcars are linked together.

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

SECTION 1.4 PolyNomiAls feet. Figure 1. A = s 2 = (2x) 2 = 4x 2 A = 2 (2x) 3 _ 2 = 1 _ = 3 _. A = lw = x 1. = x

Summer Prep Packet for students entering Algebra 2

Answer Key. Solve each equation x - 9 = (n + 2) = b - 6 = -3b + 48

y ax bx c OR 0 then either a = 0 OR b = 0 Steps: 1) if already factored, set each factor in ( ) = 0 and solve

LT1: Adding and Subtracting Polynomials. *When subtracting polynomials, distribute the negative to the second parentheses. Then combine like terms.

LESSON 7.2 FACTORING POLYNOMIALS II

Rising 8th Grade Math. Algebra 1 Summer Review Packet

UNIT 2 FACTORING. M2 Ch 11 all

Maintaining Mathematical Proficiency

Name: Chapter 7: Exponents and Polynomials

Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics

2 P a g e. Essential Questions:

Algebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials

Multiplication of Polynomials

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

= The algebraic expression is 3x 2 x The algebraic expression is x 2 + x. 3. The algebraic expression is x 2 2x.

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

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

Unit 3 Factors & Products

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

Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.

8-1: Adding and Subtracting Polynomials

Algebra I. Polynomials.

Adding and Subtracting Polynomials

Polynomials Practice Test

Unit 13: Polynomials and Exponents

Math 10 - Unit 5 Final Review - Polynomials

Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms

POLYNOMIAL: A polynomial is a or the

Math 1 Variable Manipulation Part 6 Polynomials

Algebra I Polynomials

Collecting Like Terms

( ) Chapter 7 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3.

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

Quadratic Expressions and Equations

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

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

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

Prerequisites. Introduction CHAPTER OUTLINE

EX: Simplify the expression. EX: Simplify the expression. EX: Simplify the expression

Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents

Quadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.

Slide 1 / 200. Quadratic Functions

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

Get Ready. 6. Expand using the distributive property. a) 6m(2m 4) b) 8xy(2x y) c) 6a 2 ( 3a + 4ab) d) 2a(b 2 6ab + 7)

Using the binomial square thm. rewrite the following trinomial into an equivalent expression (The square of a binomial) 25c 2 70cd + 49d 2

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2

6-5 Multiplying Polynomials

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

LESSON 9.1 ROOTS AND RADICALS

Note: A file Algebra Unit 09 Practice X Patterns can be useful to prepare students to quickly find sum and product.

When factoring, we ALWAYS start with the (unless it s 1).

Algebra I. Exponents and Polynomials. Name

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

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

ACTIVITY: Classifying Polynomials Using Algebra Tiles

Completing the Square

Lesson 3: Polynomials and Exponents, Part 1

Day 131 Practice. What Can You Do With Polynomials?

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

CHAPTER 1 POLYNOMIALS

5.1 Monomials. Algebra 2

Name Class Date. Multiplying Two Binomials Using Algebra Tiles

Sections 7.2, 7.3, 4.1

Algebra 1 Unit 6B Factoring

Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions

Sections 7.1, 7.2: Sums, differences, products of polynomials CHAPTER 7: POLYNOMIALS

Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o

Math 096--Quadratic Formula page 1

= 9 = x + 8 = = -5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives:

Unit 3A: Factoring & Solving Quadratic Equations After completion of this unit, you will be able to

Algebra 1B Final Review

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

Find a common monomial factor. = 2y 3 (y + 3)(y 3) Difference of two squares

Unit 5 Quadratic Expressions and Equations

Factoring x 2 + bx + c

Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers

SUMMER ASSIGNMENT FOR ALGEBRA II/TRIGONOMETRY

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

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

Chapter 8 Class Notes 8-A1 (Lessons 8-1&8-2) Monomials and Factoring p Prime Factorization: a whole number expressed as the of factors.

Quadratic Functions and Equations

Math 2 Variable Manipulation Part 3 Polynomials A

Factoring Polynomials Using the GCF

Chapter 6. Polynomials

Transcription:

Chapter 8 Polynomials and Factoring

8.1 Add and Subtract Polynomials

Monomial A. EX: Degree of a monomial the of all of the of the EX: 4x 2 y

Polynomial A or EX: Degree of a polynomial the of its terms EX: 2x 2 + 5x + 7

Leading coefficient the coefficient of the when the polynomial is written in EX: 2x 2 + 5x + 7

EX: Write the polynomial so that the exponents decrease from left to right. Identify the degree and the leading coefficient. 7 5y 3-5 + 2x 2 + x 3 7x

Binomial a polynomial with EX: Trinomial a polynomial with EX:

To add polynomials - Add REMEMBER: You can only add if the AND the are the same. EX: Find the sum. (6a 2 4) + (2a 2 9)

EX: Find the sum. (5x 3 + 4x 2x) + (4x 2 + 3x 3 6)

To subtract polynomials - Distribute the then. Make sure to when distributing the negative. EX: Find the difference. (4n 2 + 5) (-2n 2 + 2n 4)

EX: Find the difference. (4x 2 7x) (5x 2 + 4x 9)

EX: Write a polynomial that represents the perimeter of the figure. All sides added up.

EX: Major League Baseball teams are divided into two leagues. During the period 1995-2001, the attendance (in thousands) at National (N) and American (A) games can be modeled by: N = -488t 2 + 5430t + 24,700 A = -318t 2 + 3040t + 25,600 Where t is the number of years since 1995. About how many total people attended games in 2001?

8.2 Multiplying Polynomials

To multiply polynomials: everything in the to everything in the. REMEMBER: When you like bases, the exponents.

EX: Find the product. x(2x 3-7x 2 + 4) (x 2)(x 2 +2x + 1)

(3y 2 y + 5)(2y 3)

FOIL Method When multiplying, you can use the FOIL Method. The FOIL Method is the but in a. Multiply: Firsts Outers Inners Lasts

FOIL Method

EX: Find the product. (4b 5)(b 2) (6n 1)(n + 5)

(2x + 3y)2

EX: Simplify the expression. -3x 2 (x+ 11) (4x 5)(3x 2)

EX: Write a polynomial that represents the area of the shaded region.

EX: You are planning to build a walkway that surrounds a rectangular garden, as shown. The width of the walkway is the same on every side. Write a polynomial that represents the combined area of the garden and the walkway. Find the combined area when the width of the walkway is 4 feet. 10 ft x 9 ft

8.4 Solve Polynomial Equations in Factored Form

Factoring To factor a polynomial To Factor a Polynomial: Step 1: Look for a A monomial that can be of in the polynomial Write the common monomial and by what is left over

EX: Factor out the greatest common monomial factor. 12x + 42y 4x 4 + 24x 3

15n 3 25n 8a 2 b 6ab 2 + 4ab

Zero-Product Property If ab = 0, then. This property is used to solve an equation when and the other side is a. EX: (x-3)(2x + 7) = 0

To solve an equation by factoring: 1) Put the equation in set equal to. EX: 2). 3) Set each factor and.

EX: Solve the equation. (2y + 5)(7y 5)= 0

a 2 + 5a = 0-28m 2 = 8m

Vertical Motion Model Models the of a An object thrown in the air with only the acting on it. http://phet.colorado.edu/sims/projectilemotion/projectile-motion_en.html

Vertical Motion Model h = -16t 2 + vt + s h = of the object (in feet) t = the object is in the air (in seconds) v = (in feet per second) s = of the object (in feet)

EX: A dolphin jumped out of the water with an initial vertical velocity of 32 feet per second. After how many seconds did the dolphin enter the water? After 1 second, how high was the dolphin above the water?

8.5 Factor x 2 + bx + c

Review: Multiplying results in a. EX: (3x + 2)(x 4) Therefore, we will.

To factor a polynomial: Step 1: Look for a. Step 2: If you have a, factor it into. The 1 st terms in each binomial must to get the 1 st term in the trinomial. The 2 nd term in each binomial must to get the coefficient in the trinomial and must to get the in the trinomial. NOTE: Pay attention to.

NOTEBOOK EXAMPLE #1 EX: Factor the trinomial. x 2 + 3x + 2 t 2 + 9t + 14 x 2 4x + 3 t 2 8t + 12 m 2 + m 20 w 2 + 6w 16 x 2 4xy + 4y 2 m 2 mn 42n 2

NOTEBOOK EXAMPLE #2 EX: Solve the equation (by factoring). x 2 2x = 24 x 2 2x 8 = 7 s(s + 1) = 72

EX: Find the dimensions of the rectangle. 8.5 Ex #43

EX: You are designing a flag for the SMCC football team with the dimensions shown. The flag requires 80 square feet of fabric. Find the width w of the flag. SMCC

8.6 Factor ax 2 + bx + c

Steps to Factoring: 1) Look for a. If the leading coefficient is, factor out a. 2) If you have a, factor it into.

Test to factor a trinomial when the leading coefficient is : In your two binomials: The 1 st terms must to get the in the trinomial. The 2 nd terms must to get the in the trinomial. plus must equal the in the trinomial.

NOTEBOOK EXAMPLE #3 EX: Factor the trinomial. 3x 2 + 8x + 4 4x 2 9x + 5 2x 2 + 13x 7-2x 2 5x 3-5m 2 + 6m 1-4n 2 16n 15

NOTEBOOK EXAMPLE #4 EX: Solve the equation (by factoring). 8x 2 2x = 3 b(20b 3) = 2 6x 2 15x = 99

EX: In a shot put event, an athlete throws the shot put from an initial height of 6 feet and with an initial vertical velocity of 29 feet per second. After how many seconds does the shot put hit the ground?

EX: The length of a rectangle is 7 inches more than 5 times its width. The area of the rectangle is 6 square inches. What is the width?

8.7 Factoring Special Products

Steps to Factoring: 1). 2) Trinomial into. If the two binomials are the, its called a. 3) If you have a : Factor it into with the but. EX: x 2 9

NOTEBOOK EXAMPLE #5 EX: Factor the polynomial. y 2 16 121x 2 144 x 2 81y 2 8 18x 2-4x 2 + 64 x 2 12x + 36 4x 2 + 4xy + y 2-3y 2 + 36y 108

NOTEBOOK EXAMPLE #6 EX: Solve the equation. 4x 2 36 = 0-8k 2 = -98 y 2-5 3 y = -25 36

EX: A window washer drops a wet sponge from a height of 64 feet. After how many seconds does the sponge land on the ground?

8.8 Factor Polynomials Completely

Steps to Factoring: 1). 2) Trinomial into. 3) Difference of square. Same, different. 4) If you have a, factor by : Group the polynomial in. Factor out a from each. Factor out a from each pair.

NOTEBOOK EXAMPLE #7 EX: Factor. x(x- 2) + (x 2) 6z(z 4) 7(z 4) 12(z 1) 5z 2 (1 z) a 3 + 3a 2 + a + 3 a 3 + 13a 2 5a 65

NOTEBOOK EXAMPLE #8 EX: Factor completely. 3x 3 12x 2y 3 12y 2 + 18y 7a 3 b 3 63ab 3

NOTEBOOK EXAMPLE #9 EX: Solve the equation. w 3 8w 2 + 16w = 0 x 4 25x 2 = 0 4y 3 7y 2 = 16y - 28

EX: A terrarium has a volume of 4608 cubic inches. Its length is more than 10 inches. Its dimensions are shown. Find the length, width, and height of the terrarium.

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback