# How To Factor By Gcf In Algebra 1.5

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1 7-2 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz Algebra 1

2 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 5 13p

3 Objective Factor polynomials by using the greatest common factor.

4 Recall that the Distributive Property states that ab + ac =a(b + c). The Distributive Property allows you to factor out the GCF of the terms in a polynomial to write a factored form of the polynomial. A polynomial is in its factored form when it is written as a product of monomials and polynomials that cannot be factored further. The polynomial 2(3x 4x) is not fully factored because the terms in the parentheses have a common factor of x.

5 Example 1A: Factoring by Using the GCF Factor each polynomial. Check your answer. 2x 2 4 2x 2 = 2 x x 4 = 2 2 Find the GCF. 2 2x 2 (2 2) 2(x 2 2) Check 2(x 2 2) 2x 2 4 The GCF of 2x 2 and 4 is 2. Write terms as products using the GCF as a factor. Use the Distributive Property to factor out the GCF. Multiply to check your answer. The product is the original polynomial.

6 Writing Math Aligning common factors can help you find the greatest common factor of two or more terms.

7 Example 1B: Factoring by Using the GCF Factor each polynomial. Check your answer. 8x 3 4x 2 16x 8x 3 = x x x 4x 2 = 2 2 x x 16x = x 2 2 x = 4x 2x 2 (4x) x(4x) 4(4x) 4x(2x 2 x 4) Check 4x(2x 2 x 4) 8x 3 4x 2 16x Find the GCF. The GCF of 8x 3, 4x 2, and 16x is 4x. Write terms as products using the GCF as a factor. Use the Distributive Property to factor out the GCF. Multiply to check your answer. The product is the original polynomials.

8 Example 1C: Factoring by Using the GCF Factor each polynomial. Check your answer. 14x 12x 2 1(14x + 12x 2 ) Both coefficients are 14x = 2 7 x negative. Factor out 1. 12x 2 = x x Find the GCF. 2 x = 2x The GCF of 14x and 12x 2 is 2x. 1[7(2x) + 6x(2x)] Write each term as a product using the GCF. 1[2x(7 + 6x)] Use the Distributive Property to factor out the GCF. 2x(7 + 6x)

9 Example 1C: Continued Factor each polynomial. Check your answer. 14x 12x 2 Check 2x(7 + 6x) 14x 12x 2 Multiply to check your answer. The product is the original polynomial.

10 Caution! When you factor out 1 as the first step, be sure to include it in all the other steps as well.

11 Example 1D: Factoring by Using the GCF Factor each polynomial. Check your answer. 3x 3 + 2x x 3 = 3 x x x 2x 2 = 2 x x 10 = 2 5 3x 3 + 2x 2 10 Find the GCF. There are no common factors other than 1. The polynomial cannot be factored further.

12 Check It Out! Example 1a Factor each polynomial. Check your answer. 5b + 9b 3 5b = 5 b 9b = 3 3 b b b b 5(b) + 9b 2 (b) b(5 + 9b 2 ) Check b(5 + 9b 2 ) 5b + 9b 3 Find the GCF. The GCF of 5b and 9b 3 is b. Write terms as products using the GCF as a factor. Use the Distributive Property to factor out the GCF. Multiply to check your answer. The product is the original polynomial.

13 Check It Out! Example 1b Factor each polynomial. Check your answer. 9d d 2 = 3 3 d d Find the GCF. 8 2 = d There are no common factors other than 1. The polynomial cannot be factored further.

14 Check It Out! Example 1c Factor each polynomial. Check your answer. 18y 3 7y 2 1(18y 3 + 7y 2 ) 18y 3 = y y y Both coefficients are negative. Factor out 1. 7y 2 = 7 y y Find the GCF. y y = y 2 The GCF of 18y 3 and 7y 2 is y 2. Write each term as a product 1[18y(y 2 ) + 7(y 2 )] 1[y 2 (18y + 7)] y 2 (18y + 7) using the GCF. Use the Distributive Property to factor out the GCF..

15 Check It Out! Example 1d Factor each polynomial. Check your answer. 8x 4 + 4x 3 2x 2 8x 4 = x x x x 4x 3 = 2 2 x x x 2x 2 = 2 x x Find the GCF. 2 x x = 2x 2 4x 2 (2x 2 ) + 2x(2x 2 ) 1(2x 2 ) 2x 2 (4x 2 + 2x 1) Check 2x 2 (4x 2 + 2x 1) 8x 4 + 4x 3 2x 2 The GCF of 8x 4, 4x 3 and 2x 2 is 2x 2. Write terms as products using the GCF as a factor. Use the Distributive Property to factor out the GCF. Multiply to check your answer. The product is the original polynomial.

16 To write expressions for the length and width of a rectangle with area expressed by a polynomial, you need to write the polynomial as a product. You can write a polynomial as a product by factoring it.

17 Example 2: Application The area of a court for the game squash is (9x 2 + 6x) square meters. Factor this polynomial to find possible expressions for the dimensions of the squash court. A = 9x 2 + 6x The GCF of 9x 2 and 6x is 3x. = 3x(3x) + 2(3x) = 3x(3x + 2) Write each term as a product using the GCF as a factor. Use the Distributive Property to factor out the GCF. Possible expressions for the dimensions of the squash court are 3x m and (3x + 2) m.

18 Check It Out! Example 2 What if? The area of the solar panel on another calculator is (2x 2 + 4x) cm 2. Factor this polynomial to find possible expressions for the dimensions of the solar panel. A = 2x 2 + 4x The GCF of 2x 2 and 4x is 2x. = x(2x) + 2(2x) = 2x(x + 2) Write each term as a product using the GCF as a factor. Use the Distributive Property to factor out the GCF. Possible expressions for the dimensions of the solar panel are 2x cm, and (x + 2) cm.

19 Sometimes the GCF of terms is a binomial. This GCF is called a common binomial factor. You factor out a common binomial factor the same way you factor out a monomial factor.

20 Example 3: Factoring Out a Common Binomial Factor Factor each expression. A. 5(x + 2) + 3x(x + 2) 5(x + 2) + 3x(x + 2) (x + 2)(5 + 3x) B. 2b(b 2 + 1)+ (b 2 + 1) 2b(b 2 + 1) + (b 2 + 1) 2b(b 2 + 1) + 1(b 2 + 1) (b 2 + 1)( 2b + 1) The terms have a common binomial factor of (x + 2). Factor out (x + 2). The terms have a common binomial factor of (b 2 + 1). (b 2 + 1) = 1(b 2 + 1) Factor out (b 2 + 1).

21 Example 3: Factoring Out a Common Binomial Factor Factor each expression. C. 4z(z 2 7) + 9(2z 3 + 1) There are no common 4z(z 2 7) + 9(2z 3 + 1) factors. The expression cannot be factored.

22 Factor each expression. Check It Out! Example 3 a. 4s(s + 6) 5(s + 6) 4s(s + 6) 5(s + 6) The terms have a common binomial factor of (s + 6). (4s 5)(s + 6) Factor out (s + 6). b. 7x(2x + 3) + (2x + 3) 7x(2x + 3) + (2x + 3) 7x(2x + 3) + 1(2x + 3) (2x + 3)(7x + 1) The terms have a common binomial factor of (2x + 3). (2x + 3) = 1(2x + 3) Factor out (2x + 3).

23 Check It Out! Example 3 : Continued Factor each expression. c. 3x(y + 4) 2y(x + 4) 3x(y + 4) 2y(x + 4) The expression cannot be factored. There are no common factors. d. 5x(5x 2) 2(5x 2) 5x(5x 2) 2(5x 2) (5x 2)(5x 2) (5x 2) 2 The terms have a common binomial factor of (5x 2 ). (5x 2)(5x 2) = (5x 2) 2

24 You may be able to factor a polynomial by grouping. When a polynomial has four terms, you can make two groups and factor out the GCF from each group.

25 Example 4A: Factoring by Grouping Factor each polynomial by grouping. Check your answer. 6h 4 4h h 8 (6h 4 4h 3 ) + (12h 8) 2h 3 (3h 2) + 4(3h 2) 2h 3 (3h 2) + 4(3h 2) (3h 2)(2h 3 + 4) Group terms that have a common number or variable as a factor. Factor out the GCF of each group. (3h 2) is another common factor. Factor out (3h 2).

26 Example 4A Continued Factor each polynomial by grouping. Check your answer. Check (3h 2)(2h 3 + 4) 3h(2h 3 ) + 3h(4) 2(2h 3 ) 2(4) Multiply to check your solution. 6h h 4h 3 8 6h 4 4h h 8 The product is the original polynomial.

27 Example 4B: Factoring by Grouping Factor each polynomial by grouping. Check your answer. 5y 4 15y 3 + y 2 3y (5y 4 15y 3 ) + (y 2 3y) 5y 3 (y 3) + y(y 3) 5y 3 (y 3) + y(y 3) (y 3)(5y 3 + y) Group terms. Factor out the GCF of each group. (y 3) is a common factor. Factor out (y 3).

28 Example 4B Continued Factor each polynomial by grouping. Check your answer. 5y 4 15y 3 + y 2 3y Check (y 3)(5y 3 + y) y(5y 3 ) + y(y) 3(5y 3 ) 3(y) 5y 4 + y 2 15y 3 3y Multiply to check your solution. 5y 4 15y 3 + y 2 3y The product is the original polynomial.

29 Check It Out! Example 4a Factor each polynomial by grouping. Check your answer. 6b 3 + 8b 2 + 9b + 12 (6b 3 + 8b 2 ) + (9b + 12) 2b 2 (3b + 4) + 3(3b + 4) 2b 2 (3b + 4) + 3(3b + 4) (3b + 4)(2b 2 + 3) Group terms. Factor out the GCF of each group. (3b + 4) is a common factor. Factor out (3b + 4).

30 Check It Out! Example 4a Continued Factor each polynomial by grouping. Check your answer. 6b 3 + 8b 2 + 9b + 12 Check (3b + 4)(2b 2 + 3) Multiply to check your solution. 3b(2b 2 ) + 3b(3)+ (4)(2b 2 ) + (4)(3) 6b 3 + 9b+ 8b b 3 + 8b 2 + 9b + 12 The product is the original polynomial.

31 Check It Out! Example 4b Factor each polynomial by grouping. Check your answer. 4r r + r (4r r) + (r 2 + 6) 4r(r 2 + 6) + 1(r 2 + 6) 4r(r 2 + 6) + 1(r 2 + 6) (r 2 + 6)(4r + 1) Group terms. Factor out the GCF of each group. (r 2 + 6) is a common factor. Factor out (r 2 + 6).

32 Check It Out! Example 4b Continued Factor each polynomial by grouping. Check your answer. Check (4r + 1)(r 2 + 6) 4r(r 2 ) + 4r(6) +1(r 2 ) + 1(6) 4r r +r r r + r Multiply to check your solution. The product is the original polynomial.

33 Helpful Hint If two quantities are opposites, their sum is 0. (5 x) + (x 5) 5 x + x 5 x + x

34 Recognizing opposite binomials can help you factor polynomials. The binomials (5 x) and (x 5) are opposites. Notice (5 x) can be written as 1(x 5). 1(x 5) = ( 1)(x) + ( 1)( 5) = x + 5 = 5 x So, (5 x) = 1(x 5) Distributive Property. Simplify. Commutative Property of Addition.

35 Example 5: Factoring with Opposites Factor 2x 3 12x x by grouping. 2x 3 12x x (2x 3 12x 2 ) + (18 3x) 2x 2 (x 6) + 3(6 x) 2x 2 (x 6) + 3( 1)(x 6) 2x 2 (x 6) 3(x 6) (x 6)(2x 2 3) Group terms. Factor out the GCF of each group. Write (6 x) as 1(x 6). Simplify. (x 6) is a common factor. Factor out (x 6).

36 Check It Out! Example 5a Factor each polynomial by grouping. 15x 2 10x 3 + 8x 12 (15x 2 10x 3 ) + (8x 12) 5x 2 (3 2x) + 4(2x 3) 5x 2 (3 2x) + 4( 1)(3 2x) 5x 2 (3 2x) 4(3 2x) (3 2x)(5x 2 4) Group terms. Factor out the GCF of each group. Write (2x 3) as 1(3 2x). Simplify. (3 2x) is a common factor. Factor out (3 2x).

37 Check It Out! Example 5b Factor each polynomial by grouping. 8y 8 x + xy (8y 8) + ( x + xy) 8(y 1)+ (x)( 1 + y) 8(y 1)+ (x)(y 1) (y 1)(8 + x) Group terms. Factor out the GCF of each group. (y 1) is a common factor. Factor out (y 1).

38 Lesson Quiz: Part I Factor each polynomial. Check your answer x + 20x 3 4x(4 + 5x 2 ) 2. 4m 4 12m 2 + 8m 4m(m 3 3m + 2) Factor each expression. 3. 7k(k 3) + 4(k 3) (k 3)(7k + 4) 4. 3y(2y + 3) 5(2y + 3) (2y + 3)(3y 5)

39 Lesson Quiz: Part II Factor each polynomial by grouping. Check your answer. 5. 2x 3 + x 2 6x 3 (2x + 1)(x 2 3) 6. 7p 4 2p p 18 (7p 2)(p 3 + 9) 7. A rocket is fired vertically into the air at 40 m/s. The expression 5t t + 20 gives the rocket s height after t seconds. Factor this expression. 5(t 2 8t 4)

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### MATH 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

### Factoring - Greatest Common Factor 6.1 Factoring - Greatest Common Factor Objective: Find the greatest common factor of a polynomial and factor it out of the expression. The opposite of multiplying polynomials together is factoring polynomials.

### GCF/ Factor by Grouping (Student notes) GCF/ Factor by Grouping (Student notes) Factoring is to write an expression as a product of factors. For example, we can write 10 as (5)(2), where 5 and 2 are called factors of 10. We can also do this

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### called and explain why it cannot be factored with algebra tiles? and explain why it cannot be factored with algebra tiles? Factoring Reporting Category Topic Expressions and Operations Factoring polynomials Primary SOL A.2c The student will perform operations on polynomials, including factoring completely first- and second-degree

### 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,

### 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

### FACTORING 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".

### CAHSEE on Target UC Davis, School and University Partnerships UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

### 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

### 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

### 5 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 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

### 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

### 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

### Chris Yuen. Algebra 1 Factoring. Early High School 8-10 Time Span: 5 instructional days 1 Chris Yuen Algebra 1 Factoring Early High School 8-10 Time Span: 5 instructional days Materials: Algebra Tiles and TI-83 Plus Calculator. AMSCO Math A Chapter 18 Factoring. All mathematics material and

### FACTORING 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

### A 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...

### 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?

### 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,...} 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

### Chapter 3 Section 6 Lesson Polynomials Chapter Section 6 Lesson Polynomials Introduction This lesson introduces polynomials and like terms. As we learned earlier, a monomial is a constant, a variable, or the product of constants and variables.

### In 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,

### Big Bend Community College. Beginning Algebra MPC 095. Lab Notebook Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond

### 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 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract

### A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply 1 Summary DEFINITION/PROCEDURE EXAMPLE REFERENCE From Arithmetic to Algebra Section 1.1 Addition x y means the sum of x and y or x plus y. Some other words The sum of x and 5 is x 5. indicating addition EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an