# 15.1 Factoring Polynomials

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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 2-5y 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 2-5y 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 2-12 y x x 3-2 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)(7-2x) 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 3-9a a - 15 (12a 3-9a 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 3-9a 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 3-15x x ( 3x 3-15x 2 ) + (10-2x) 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 2-2) Factor out (x - 5). Check: (x - 5)( 3x 2-2) Multiply using FOIL. x( 3x 2 ) - x(2) - 5( 3x 2 ) - 5(-2) 3x 3-2x - 15x x 3-15x 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 2-10x 3 + 8x y x + xy n 6-18n 5-56n t 4-48t 3-3t 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 2-14y 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 3-7y 2 - y -y( y ) 5. 9d 2-18 ( d 2 - ) 6. 6 x 4-2x 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 3-40x2 + 14x n 5-2n4 + 7n2-7n Factor each polynomial. (Example 5) r 2-6r r q 2-21q + 6-4q (2 r 2 - ) + ( ) ( ) + ( ) ( - 3) + ( ) 7q( ) + 2( ) 2r(r - 3) + 4 ( ) 7q( ) + 2 ( ) ( )( ) ( )( ) ( )( )? 18. 6c c 2-5c x 3-27x x ESSENTIAL QUESTION CHECK-IN 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 3-35r2 + 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 3-2x + 2 x 2-4). 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

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### 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, 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

### 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?

### 1.4. 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.

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

### 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

### A. Factoring out the Greatest Common Factor. DETAILED SOLUTIONS AND CONCEPTS - FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!

### 1.1 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)

### 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...

### Chapter 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

### Factoring. 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 9-2 through 9-6 Factor polynomials.

### Factoring 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

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### SPECIAL PRODUCTS AND FACTORS SPECIAL PRODUCTS AND FACTORS I. INTRODUCTION AND FOCUS QUESTIONS http://dmciresidences.com/home/20/0/ cedar-crest-condominiums/ http://frontiernerds.com/metal-box http://mazharalticonstruction.blogspot. 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 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

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### Factoring Trinomials of the Form x 2 bx c 4.2 Factoring Trinomials of the Form x 2 bx c 4.2 OBJECTIVES 1. Factor a trinomial of the form x 2 bx c 2. Factor a trinomial containing a common factor NOTE The process used to factor here is frequently

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### 6.4 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

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

### Factoring. 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

### Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic Sect 6.1 - Greatest Common Factor and Factoring by Grouping Our goal in this chapter is to solve non-linear equations by breaking them down into a series of linear equations that we can solve. To do this, 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