15.1 Factoring Polynomials


 May Hoover
 5 years ago
 Views:
Transcription
1 LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE ACTIVITY Factoring and Greatest Common Factor Factors that are shared by two or more whole numbers are called common factors. The greatest of these common factors is called the greatest common factor, or GCF. The greatest common factor is 4. Use the greatest common factor (GCF) and the Distributive Property to factor the expression 30x A Write out the prime factors of each term. 30x + 18 = 2 x + 2 B Circle the common factors. 30x + 18 = 2 x + 2 C Write the expression as the product of the GCF and a sum. 30x + 18 = ( ) ( x + ) REFLECT 1. Will you get a completely factored expression if you factor out a common factor that is not the GCF? Explain. 2. Is the expression 2(3x  4x) completely factored? Explain. Lesson
2 Greatest Common Factor of Monomials To find the GCF of monomials, factor each coefficient and write all powers of variables as products. Then find the product of the common factors. Math On the Spot EXAMPLE 1 Find the GCF of each pair of monomials. A 3 x 3 and 6 x 2 3x 3 = 3 x x x 6x 2 = 2 3 x x Factor each coefficient and write powers as products. Find the common factors. 3 x x Find the product of the common factors. The GCF of 3x 3 and 6x 2 is 3x 2. B 4x 2 and 5y 3 Math Talk Mathematical Practices Does factoring an expression change its value? 4x 2 = 2 2 x x 5y 3 = 5 y y y Factor each coefficient and write powers as products. Since there are no common factors other than 1, the GCF of 4x 2 and 5y 3 is 1. REFLECT 3. Analyze Relationships If two terms contain the same variable raised to different powers, to what power will the variable be raised in the GCF? 4. Can the GCF of two positive numbers be greater than both numbers? Explain. YOUR TURN Find the GCF of each pair of monomials g 2 and 27g a 6 and 9b g 4 and 45g ab and 16bc 524 Unit 4
3 Factoring by Using the GCF Remember that the Distributive Property states that ab + ac = a(b + c). Use the Distributive Property to factor out the GCF of the terms in a polynomial to write the polynomial in factored form. EXAMPLE 2 Math On the Spot Factor each polynomial. Check your answer. A 10y y 25y 2y 2 (5y) + 4y(5y)  1(5y) The GCF is 5y. 5y( 2y 2 + 4y  1) Use the Distributive Property. My Notes Check: 5y( 2y 2 + 4y  1) 10y y 25y The product is the original polynomial. B 12x  8x 2 Both coefficients are negative. 1(12x + 8x 2 ) Factor out 1. 1[3(4x) + 2x(4x)] The GCF of 12x and 8x 2 is 4x. 1[4x(3 + 2x)] 1(4x)(3 + 2x) Use the Distributive Property. Use the Associative Property. 4x(3 + 2x) Check: 4x(3 + 2x) = 12x  8x 2 The product is the original polynomial. REFLECT 9. Can the polynomial 5x be factored? Explain. YOUR TURN Factor each polynomial. Check your answer y 212 y x x 32 x 2 Lesson
4 Math On the Spot Factoring Out a Common Binomial Factor Sometimes the GCF of the terms in an expression is a binomial. Such a GCF is called a common binomial factor. You factor out a common binomial factor the same way you factor out a monomial factor. EXAMPLE 3 Factor each expression. My Notes A 7(x  3)  2x(x  3) 7(x 3)  2x(x 3) (x  3) is a common binomial factor. (x 3)(72x) Factor out (x  3). B t(t2 + 4) + (t2 + 4) t( t 2 + 4) + ( t 2 + 4) ( t 2 + 4) is a common binomial factor. t( t 2 + 4) + 1( t 2 + 4) ( t 2 + 4) = 1( t 2 + 4) ( t 2 + 4)(t + 1) Factor out ( t 2 + 4). C 5x(x + 3)  4(3 + x) 5x(x + 3)  4(3 + x) 5x(x + 3)  4(x + 3) (3 + x) = (x + 3), so (x + 3) is a common binomial factor. (x + 3)(5x  4) Factor out (x + 3). D 3x2 (x + 2) + 4(x  7) 3x 2 (x + 2) + 4(x  7) There are no common factors. The expression cannot be factored. YOUR TURN Factor each expression, if possible x(2x + 3) + (2x + 3) x(x + 2) + 9(x + 2) 14. 7(3t  2) + 2t2 (2t  3) 15. 5t(t + 6)  8(6 + t) 526 Unit 4
5 Factoring by Grouping Some polynomials can be factored by grouping. When a polynomial has four terms, you may be able to make two groups and factor the GCF from each. EXAMPLE 4 Factor each polynomial by grouping. Check your answer. Math On the Spot A 12a 39a a  15 (12a 39a 2 ) + (20a  15) 3a 2 (4a  3) + 5(4a  3) 3a 2 (4a  3) + 5(4a  3) Group terms that have a common number or variable as a factor. Factor out the GCF of each group. (4a  3) is a common factor. (4a  3)( 3a 2 + 5) Factor out (4a  3). Check: (4a  3)( 3a 2 + 5) Multiply using FOIL. 4a( 3a 2 ) + 4a(5)  3( 3a 2 )  3(5) 12a a  9a a 39a a  15 The product is the original polynomial. B 2g g 3 + g + 5 ( 2g g 3 ) + (g + 5) Group terms. 2g 3 (g + 5) + 1(g + 5) 2g 3 (g + 5) + 1(g + 5) Factor out the GCF of each group. (g + 5) is a common factor. (g + 5)( 2g 3 + 1) Factor out (g + 5). Check: (g + 5)( 2g 3 + 1) Multiply using FOIL. g( 2g 3 ) + g(1) + 5( 2g 3 ) + 5(1) 2g 4 + g + 10g g g 3 + g + 5 The product is the original polynomial. YOUR TURN Factor each polynomial. Check your answer b 3 + 8b 2 + 9b r r + r Lesson
6 Factoring with Opposites Recognizing opposite binomials can help you factor polynomials. The binomials (5  x) and (x  5) are opposites, because (5  x) = 1(x  5). Math On the Spot EXAMPLE 5 Factor the polynomial by grouping and using opposites. Check your answer. My Notes 3x 315x x ( 3x 315x 2 ) + (102x) Group terms. 3x 2 (x  5) + 2(5  x) Factor out the GCF of each group. 3x 2 (x 5) + 2( 1)(x 5) Write (5  x) as 1(x  5). 3x 2 (x 5)  2(x 5) Simplify. (x 5)( 3x 22) Factor out (x  5). Check: (x  5)( 3x 22) Multiply using FOIL. x( 3x 2 )  x(2)  5( 3x 2 )  5(2) 3x 32x  15x x 315x x The product is the original polynomial. REFLECT 18. Critique Reasoning Inara thinks that the opposite of (a  b) is (a + b), since addition and subtraction are opposites. Is she correct? Explain. YOUR TURN Factor each polynomial. Check your answer x 210x 3 + 8x y x + xy n 618n 556n t 448t 33t Unit 4
7 Guided Practice Write the expression as a product of the greatest common factor and a sum. (Explore Activity) 1. 15y y a. Write out the prime factors of each term. 15y3 + 20y = 3 y + 2 y b. Circle the common factors. 15y3 + 20y = 3 y + 2 y c. Write the product of the GCF and a sum. 15y3 + 20y = ( )( y2 + ) Find the GCF of each pair of monomials. (Example 1) 2. 9s and 63s 3 9s = y y 214y 3 = 63s 3 = y 2 = The GCF of 9s and 63s 3 is. The GCF of 14y 3 and 28y 2 is. Factor each polynomial. (Example 2) y 37y 2  y y( y ) 5. 9d 218 ( d 2  ) 6. 6 x 42x x t Factor each expression. (Example 3) 8. 4s(s + 6)  5(s + 6) 9. 3(2 + b) + 4b(b + 2) ( )(s + 6) ( )( ) 10. (6z)(z + 8) + (z + 8) 11. 8w(5  w) + 3(w  5) Lesson
8 Factor each polynomial. (Example 4) 12. 9x x 2 + x m 3 + 4m 2 + 6m + 12 ( 9x 3 + ) + ( ) ( + 4m 2 ) + ( ) (x + ) + ( ) (m + ) + (m + ) ( )( ) (m + )( ) 2(m + )( ) x 340x2 + 14x n 52n4 + 7n27n Factor each polynomial. (Example 5) r 26r r q 221q + 64q (2 r 2  ) + ( ) ( ) + ( ) (  3) + ( ) 7q( ) + 2( ) 2r(r  3) + 4 ( ) 7q( ) + 2 ( ) ( )( ) ( )( ) ( )( )? 18. 6c c 25c x 327x x ESSENTIAL QUESTION CHECKIN 20. How can you use the greatest common factor to factor polynomials? 530 Unit 4
9 Name Class Date 15.1 Independent Practice 21. Find the GCF of 64n4 and 24n 2. Factor each expression or state if it cannot be factored q p, A.SSE n n + 7n 29. After t years, the amount of money in a savings account that earns simple interest is P + Prt, where P is the starting amount and r is the yearly interest rate. Factor this expression. 30. Communicate Mathematical Ideas Explain how you can show that (x a) and (a x) are opposites b(b + 3) + 5(b + 3) 25. 4(x  3)  x (y + 2) 26. 7r 335r2 + 6r Explain how to check that a polynomial has been factored correctly. 31. The solar panel on Mandy s calculator has an area of ( 7x 2 + x) cm 2. Factor this polynomial to find possible expressions for the dimensions of the solar panel. 28. Explain the Error Billie says the factored form of 18 x 8 9 x 4 6 x 3 is 3x(6 x 7 3 x 3 2x 2 ). Explain her error and give the correct factored form. 32. A model rocket is fired vertically into the air at 320 ft/s. The expression 16t t gives the rocket s height after t seconds. Factor this expression. 33. The area of a triangle is 1_ 2 (x 32x + 2 x 24). The height h is x + 2. Write an expression for the base b of the triangle. (Hint: Area of a triangle = 1_ 2 bh) Lesson
10 34. Raspberries come in a container with a square bottom whose bottom side length is x. An expression for its volume is x 3 2x 2. Blueberries come in a container with a square bottom whose bottom side length is (x 2). An expression for its volume is x 3 4x 2 + 4x. Factor both expressions. 35. The area of a rectangle is represented by the polynomial x 2 + 3x 6x 18. a. Find possible expressions for the length and width of the rectangle. b. Use your answers from part a to find the length, width, and area of the rectangle if x = 12. FOCUS ON HIGHER ORDER THINKING Work Area 36. Critical Thinking Show two methods of factoring the expression ax  bx  ay + by. Is the result the same? 37. Explain the Error Audrey and Owen came up with two different answers when they factored the expression 3 n 3 n 2. Who was correct? Explain the error. 38. Communicating Mathematical Ideas Describe how to find the area of the figure. Show each step and write your answer in factored form. Owen Audrey 3 n 3  n 2 3n 3  n 2 n 2 (3n)  n 2 (0) n 2 (3n)  n 2 (1) n 2 (3n  0) n 2 (3n  1) 2x 2x + 6 x + 8 Image Credits: Getty Images/Photodisc 532 Unit 4
How To Factor By Gcf In Algebra 1.5
72 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Simplify. 1. 2(w + 1) 2. 3x(x 2 4) 2w + 2 3x 3 12x Find the GCF of each pair of monomials. 3. 4h 2 and 6h 2h 4. 13p and 26p
More information1.3 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 informationDefinitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).
Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32
More informationChapter R.4 Factoring Polynomials
Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationFactors and Products
CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square
More informationGreatest Common Factor (GCF) Factoring
Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication
More informationName Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
More informationFactoring Polynomials
Factoring Polynomials 8A Factoring Methods 81 Factors and Greatest Common Factors Lab Model Factoring 82 Factoring by GCF Lab Model Factorization of Trinomials 83 Factoring x 2 + bx + c 84 Factoring
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More informationWhen factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.
Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property
More informationOperations with Algebraic Expressions: Multiplication of Polynomials
Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the
More informationFactoring (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 informationIn algebra, factor by rewriting a polynomial as a product of lowerdegree polynomials
Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting
More informationFACTORING 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 informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More information6.1 The Greatest Common Factor; Factoring by Grouping
386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.
More informationFactoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai
Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationFactoring. Factoring Monomials Monomials can often be factored in more than one way.
Factoring Factoring is the reverse of multiplying. When we multiplied monomials or polynomials together, we got a new monomial or a string of monomials that were added (or subtracted) together. For example,
More informationFactoring Polynomials
Factoring Polynomials 8A Factoring Methods 81 Factors and Greatest Common Factors Lab Model Factorization by GCF 82 Factoring by GCF Lab Model Factorization of x 2 + bx + c 83 Factoring x 2 + bx + c
More informationESSENTIAL QUESTION How can you factor expressions of the form ax 2 + bx + c?
LESSON 15.3 Factoring ax 2 + bx + c A.SSE.2 Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you factor expressions of the form ax 2 + bx + c?
More information76. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content
76 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 informationMATH 90 CHAPTER 6 Name:.
MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a
More informationHow To Solve Factoring Problems
05W4801AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring
More informationIn the above, the number 19 is an example of a number because its only positive factors are one and itself.
Math 100 Greatest Common Factor and Factoring by Grouping (Review) Factoring Definition: A factor is a number, variable, monomial, or polynomial which is multiplied by another number, variable, monomial,
More informationFACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c
Tallahassee Community College 55 FACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c This kind of trinomial differs from the previous kind we have factored because the coefficient of x is no longer "1".
More information6.3 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 informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
More informationFACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1
5.7 Factoring ax 2 bx c (549) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.
More information5.1 FACTORING OUT COMMON FACTORS
C H A P T E R 5 Factoring he sport of skydiving was born in the 1930s soon after the military began using parachutes as a means of deploying troops. T Today, skydiving is a popular sport around the world.
More informationSection 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
More informationFactor Polynomials Completely
9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping
More informationFactoring Polynomials
Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationVeterans 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 informationThis is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).
This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationAlgebra 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
More informationFactoring 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 informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationFactoring 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 informationPolynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
More informationFACTORING POLYNOMIALS
296 (540) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated
More informationTool 1. Greatest Common Factor (GCF)
Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
More informationThe Greatest Common Factor; Factoring by Grouping
296 CHAPTER 5 Factoring and Applications 5.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.
More informationAIP 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 informationPOLYNOMIALS and FACTORING
POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use
More informationUsing the ac Method to Factor
4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trialanderror
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationPolynomial 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 informationFACTORING ax 2 bx c WITH a 1
296 (6 20) Chapter 6 Factoring 6.4 FACTORING a 2 b c WITH a 1 In this section The ac Method Trial and Error Factoring Completely In Section 6.3 we factored trinomials with a leading coefficient of 1. In
More informationSOL WarmUp Graphing Calculator Active
A.2a (a) Using laws of exponents to simplify monomial expressions and ratios of monomial expressions 1. Which expression is equivalent to (5x 2 )(4x 5 )? A 9x 7 B 9x 10 C 20x 7 D 20x 10 2. Which expression
More informationAlgebra 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 informationFactoring Special Polynomials
6.6 Factoring Special Polynomials 6.6 OBJECTIVES 1. Factor the difference of two squares 2. Factor the sum or difference of two cubes In this section, we will look at several special polynomials. These
More information2x 2x 2 8x. Now, let s work backwards to FACTOR. We begin by placing the terms of the polynomial inside the cells of the box. 2x 2
Activity 23 Math 40 Factoring using the BOX Team Name (optional): Your Name: Partner(s): 1. (2.) Task 1: Factoring out the greatest common factor Mini Lecture: Factoring polynomials is our focus now. Factoring
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More information5 means to write it as a product something times something instead of a sum something plus something plus something.
Intermediate algebra Class notes Factoring Introduction (section 6.1) Recall we factor 10 as 5. Factoring something means to think of it as a product! Factors versus terms: terms: things we are adding
More informationA Systematic Approach to Factoring
A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool
More informationIntroduction Assignment
PRECALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying
More informationFactoring  Grouping
6.2 Factoring  Grouping Objective: Factor polynomials with four terms using grouping. The first thing we will always do when factoring is try to factor out a GCF. This GCF is often a monomial like in
More informationBy reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.
SECTION 5.4 Special Factoring Techniques 317 5.4 Special Factoring Techniques OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationFactoring Trinomials of the Form
Section 4 6B: Factoring Trinomials of the Form A x 2 + Bx + C where A > 1 by The AC and Factor By Grouping Method Easy Trinomials: 1 x 2 + Bx + C The last section covered the topic of factoring second
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationFactoring Algebra Chapter 8B Assignment Sheet
Name: Factoring Algebra Chapter 8B Assignment Sheet Date Section Learning Targets Assignment Tues 2/17 Find the prime factorization of an integer Find the greatest common factor (GCF) for a set of monomials.
More informationFactoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1)
Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if
More information6706_PM10SB_C4_CO_pp192193.qxd 5/8/09 9:53 AM Page 192 192 NEL
92 NEL Chapter 4 Factoring Algebraic Epressions GOALS You will be able to Determine the greatest common factor in an algebraic epression and use it to write the epression as a product Recognize different
More informationPolynomials and Factoring
7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of
More informationMath 25 Activity 6: Factoring Advanced
Instructor! Math 25 Activity 6: Factoring Advanced Last week we looked at greatest common factors and the basics of factoring out the GCF. In this second activity, we will discuss factoring more difficult
More informationThe majority of college students hold credit cards. According to the Nellie May
CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationIn this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).
CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationMath 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:
Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More informationSect 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 informationA. Factoring out the Greatest Common Factor.
DETAILED SOLUTIONS AND CONCEPTS  FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
More information1.1 Practice Worksheet
Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)
More informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
More informationChapter 5. Rational Expressions
5.. Simplify Rational Expressions KYOTE Standards: CR ; CA 7 Chapter 5. Rational Expressions Definition. A rational expression is the quotient P Q of two polynomials P and Q in one or more variables, where
More informationFactoring. 472 Chapter 9 Factoring
Factoring Lesson 9 Find the prime factorizations of integers and monomials. Lesson 9 Find the greatest common factors (GCF) for sets of integers and monomials. Lessons 92 through 96 Factor polynomials.
More informationFactoring Trinomials using Algebra Tiles Student Activity
Factoring Trinomials using Algebra Tiles Student Activity Materials: Algebra Tiles (student set) Worksheet: Factoring Trinomials using Algebra Tiles Algebra Tiles: Each algebra tile kits should contain
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationSPECIAL PRODUCTS AND FACTORS
SPECIAL PRODUCTS AND FACTORS I. INTRODUCTION AND FOCUS QUESTIONS http://dmciresidences.com/home/20/0/ cedarcrestcondominiums/ http://frontiernerds.com/metalbox http://mazharalticonstruction.blogspot.
More informationRadicals  Rationalize Denominators
8. Radicals  Rationalize Denominators Objective: Rationalize the denominators of radical expressions. It is considered bad practice to have a radical in the denominator of a fraction. When this happens
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationCHAPTER 7: FACTORING POLYNOMIALS
CHAPTER 7: FACTORING POLYNOMIALS FACTOR (noun) An of two or more quantities which form a product when multiplied together. 1 can be rewritten as 3*, where 3 and are FACTORS of 1. FACTOR (verb)  To factor
More informationFactoring Trinomials of the Form x 2 bx c
4.2 Factoring Trinomials of the Form x 2 bx c 4.2 OBJECTIVES 1. Factor a trinomial of the form x 2 bx c 2. Factor a trinomial containing a common factor NOTE The process used to factor here is frequently
More informationFactoring Polynomials
Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,
More information6.4 Special Factoring Rules
6.4 Special Factoring Rules OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor a sum of cubes. By reversing the rules for multiplication
More informationSIMPLIFYING ALGEBRAIC FRACTIONS
Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is
More informationSimplifying 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 informationFactoring. Key Vocabulary
8 Factoring Find the prime factorization of integers and monomials. Factor polynomials. Use the Zero Product Property to solve equations. Key Vocabulary factored form (p. 41) perfect square trinomials
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationSect 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 nonlinear equations by breaking them down into a series of linear equations that we can solve. To do this,
More informationWentzville School District Algebra 1: Unit 8 Stage 1 Desired Results
Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Unit Title: Quadratic Expressions & Equations Course: Algebra I Unit 8  Quadratic Expressions & Equations Brief Summary of Unit: At
More information