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Unpacking the Standards Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this chapter. CC.9-12.A.SSE.2 What It Means For You Lessons 7-1, 7-2, 7-3, 7-4, 7-5, 7-6 Use the structure of an expression to identify ways to rewrite it. You will learn to factor expressions, which means you will rewrite them as a product of two or more expressions. Being able to recognize patterns will help you decide which method to use. Key Vocabulary EXAMPLE expression (expresión) A mathematical phrase that contains operations, numbers, and/or variables. Factor x 2 + 7x + 6 The algebra tiles below show that x2 + 7x + 6 = (x + 1)(x + 6). x + 6 x + 1 Factor 6x2 - 11x + 3 EXAMPLE 6x2 - 11x + 3 6x ⋅ x or 3x ⋅ 2x 1 ⋅ 3 Guess and check: (6x - 1)(x - 3) = 6x2 - 18x - x + 3 = 6x2- 19x + 3 ✗ Guess and check: (3x - 1)(2x - 3) = 6x2 - 9x - 2x + 3 = 6x2- 11x + 3 ✓ EXAMPLE Factor x 2 - 49 Use the difference of two squares pattern: Chapter 7 a2 - b2 = (a + b)(a - b) x 2- 49 = (x + 7)(x - 7) 368 Factoring Polynomials © Houghton Mifflin Harcourt Publishing Company chapter 7 × What It Means For You CC.9-12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. Key Vocabulary expression (expresión) A mathematical phrase that contains operations, numbers, and/or variables. Lessons 7-1, 7-2, 7-5, 7-6 You will learn to recognize when the terms of an expression share common factors. You will also learn to recognize when an expression fits a certain pattern based on the structure of its parts. EXAMPLE Factor Out the Greatest Common Factor 8x3 y2 + 6x2 y + 10xy = 2xy(4x2 y) + 2xy(3x) + 2xy(5) = 2xy(4x2y + 3x + 5) EXAMPLE Factor by Grouping 3x2 + 6x + 5x + 10 = 3x(x) + 3x(2) + 5(x) + 5(2) = 3x(x + 2) + 5(x + 2) = (3x + 5)(x + 2) EXAMPLE Factor a Perfect-Square Trinomial A square quilt is made up of squares sewn together in a grid pattern with a one-foot border on all sides. The side length of each square is x feet. The area in square feet is given by the expression 100x2 + 40x + 4. Write an expression for the side length of the quilt. Use a formula: Area of square = Side length squared 100x 2+ 40x + 4 = (10x + 2)2 Factor: The side length in feet is given by the expression 10x + 2. Chapter 7 369 Factoring Polynomials chapter 7 © Houghton Mifflin Harcourt Publishing Company; Photo credit: © Ruth Black/Alamy Images Key Vocabulary binomial (binomio) A polynomial with two terms. difference of two squares (diferencia de dos cuadrados) A polynomial of the form a2 - b2 , which may be written as the product (a + b)(a - b). expression (expresión) A mathematical phrase that contains operations, numbers, and/or variables. factor (factor) A number or expression that is multiplied by another number or expression to get a product. See also factoring. factoring (factorización) The process of writing a number or algebraic expression as a product. greatest common factor (monomials) (GCF) (máximo común divisor (monomios) (MCD)) The product of the greatest integer and the greatest power of each variable that divide evenly into each monomial. monomial (monomio) A number or a product of numbers and variables with whole-number exponents, or a polynomial with one term. chapter 7 perfect-square trinomial (trinomio cuadrado perfecto) A trinomial whose factored form is the square of a binomial. A perfect-square trinomial has the form a 2- 2ab + b2 = (a - b) 2or 2 2 a + 2ab + b = (a + b) 2. polynomial (polinomio) A monomial or a sum or difference of monomials. trinomial (trinomio) A polynomial with three terms. The Common Core Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. Opportunities to develop these practices are integrated throughout this program. 1. M ake sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. C onstruct viable arguments and critique the reasoning of others. 4. M odel with mathematics. 5. Use appropriate tools strategically. 6. A ttend to precision. 7. L ook for and make use of structure. 8. L ook for and express regularity in repeated reasoning Chapter 7 370 Factoring Polynomials