666-667 C13LO-825200 3/7/03 7:41 AM Page 666 Polynomials and Nonlinear Functions • Lesson 13-1 Identify and classify polynomials. Key Vocabulary • Lessons 13-2 through 13-4 Add, subtract, and • polynomial (p. 669) multiply polynomials. • degree (p. 670) • Lesson 13-5 Determine whether functions are • nonlinear function (p. 687) linear or nonlinear. • quadratic function (p. 688) • Lesson 13-6 Explore different representations • cubic function (p. 688) of quadratic and cubic functions. You have studied situations that can be modeled by linear functions. Many real-life situations, however, are not linear. These can be modeled using nonlinear functions. You will use a nonlinear function in Lesson 13-6 to determine how far a skydiver falls in 4.5 seconds. 666 Chapter 13 Polynomials and Nonlinear Functions 666-667 C13LO-865108 11/6/03 5:36 PM Page 667 Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 13. For Lesson 13-1 Monomials Determine the number of monomials in each expression. (For review, see Lesson 4-1.) 1. 2x3 2. a ϩ 4 3. 8s Ϫ 5t 1 4. x2 ϩ 3x Ϫ 1 5. ᎏᎏ 6. 9x3 ϩ 6x2 ϩ 8x Ϫ 7 t For Lesson 13-4 Distributive Property Use the Distributive Property to write each expression as an equivalent algebraic expression. (For review, see Lesson 3-1.) 7. 5(a ϩ 4) 8. 2(3y Ϫ 8) 9. Ϫ4(1 ϩ 8n) 10. 6(x ϩ 2y) 11. (9b Ϫ 9c)3 12. 5(q Ϫ 2r ϩ 3s) For Lesson 13-5 Linear Functions Determine whether each equation is linear. (For review, see Lesson 8-2.) 1 13. y ϭ x Ϫ 2 14. y ϭ x2 15. y ϭϪᎏᎏx 2 Polynomials Make this Foldable to help you organize your notes. Begin with a sheet of 11" by 17" paper. Fold Fold Again Fold the short Fold the top sides toward to the bottom. the middle. Cut Label Open. Cut along Label each the second fold to of the tabs ϩ Ϫ make four tabs. as shown. ϫ unctions F Reading and Writing As you read and study the chapter, write examples of each concept under each tab. Chapter 13 Polynomials and Nonlinear Functions 667 668-673 C13L1-865108 11/6/03 5:41 PM Page 668 Prefixes and Polynomials You can determine the meaning of many words used in mathematics if you know what the prefixes mean. In Lesson 4-1, you learned that the prefix mono means one and that a monomial is an algebraic expression with one term. Monomials Not Monomials 5 x ϩ y 2x 8n2 Ϫ n ϩ 1 y3 a3 ϩ 4a2 ϩ a Ϫ 6 The words in the table below are used in mathematics and in everyday life. They contain the prefixes bi, tri, and poly. Prefix Words bi • bisect – to divide into two congruent parts • biannual – occurring twice a year • bicycle – a vehicle with two wheels XY Z bisect tri • triangle – a figure with three sides A • triathlon – an athletic contest with three phases • trilogy – a series of three related literary works, such as films or CB books triangle poly • polyhedron – a solid with many flat surfaces • polychrome – having many colors • polygon – a figure with many sides polyhedron Reading to Learn 1. How are the words in each group of the table related? 2. What do the prefixes bi, tri, and poly mean? 3. Write the definition of binomial, trinomial, and polynomial. 4. Give an example of a binomial, a trinomial, and a polynomial. 5. RESEARCH Use the Internet or a dictionary to make a list of other words that have the prefixes bi, tri, and poly. Give the definition of each word. 668 668 Chapter 13 Polynomials and Nonlinear Functions 668-673 C13L1-825200 3/6/03 7:55 AM Page 669 Polynomials • Identify and classify polynomials. • Find the degree of a polynomial. Vocabulary are polynomials used to approximate real-world data? • polynomial Heat index is a way to describe how hot it feels outside with the • binomial temperature and humidity combined. Some examples are shown below. • trinomial • degree Temperature (˚F) Humidity (%) 80 90 100 40 79 93 110 Heat 45 80 95 115 Index 50 81 96 120 To calculate heat index, meteorologists use an expression similar to the one below. In this expression, x is the percent humidity, and y is the temperature. Ϫ42 ϩ 2x ϩ 10y Ϫ 0.2xy Ϫ 0.007x2 Ϫ 0.05y2 ϩ 0.001x2y ϩ 0.009xy2 Ϫ 0.000002x2y2 a. How many terms are in the expression for the heat index? b. What separates the terms of the expression? CLASSIFY POLYNOMIALS Recall that a monomial is a number, a variable, Study Tip or a product of numbers and/or variables. An algebraic expression that Classifying contains one or more monomials is called a polynomial. In a polynomial, Polynomials there are no terms with variables in the denominator and no terms with Be sure expressions are variables under a radical sign. written in simplest form. • x ϩ x is the same as 2x, A polynomial with two Number Polynomial Examples so the expression is a terms is called a binomial, of Terms monomial. and a polynomial with three 3 • ͙25ෆ is the same as 5, terms is called a trinomial. monomial 1 4, x, 2y ϩ Ϫ 2 ϩ so the expression is a binomial 2 x 1, a 5b, c d 2 monomial. trinomial 3 a ϩ b ϩ c, x ϩ 2x ϩ 1 The terms in a binomial or a trinomial may be added or subtracted. Example 1 Classify Polynomials Determine whether each expression is a polynomial. If it is, classify it as a monomial, binomial, or trinomial. 1 a. 2x3 ϩ 5x ϩ 7b.t Ϫ ᎏᎏ t2 This is a polynomial because The expression is not a it is the sum of three ᎏ1ᎏ polynomial because 2 has monomials. There are three t a variable in the denominator. terms, so it is a trinomial. Concept Check Is 0.5x ϩ 10 a polynomial? Explain. www.pre-alg.com/extra_examples Lesson 13-1 Polynomials 669 668-673 C13L1-825200 3/6/03 12:35 PM Page 670 DEGREES OF POLYNOMIALS The degree of a monomial is the sum of the exponents of its variables. The degree of a nonzero constant such as 6 or 10 is 0. The constant 0 has no degree. Example 2 Degree of a Monomial Find the degree of each monomial. Study Tip a. 5a b. Ϫ3x2y Degrees The variable a has degree 1, x2 has degree 2 and y has degree 1. The degree of a is 1 so the degree of 5a is 1. The degree of Ϫ3x2y is 2 ϩ 1 or 3. because a ϭ a1. A polynomial also has a degree. The degree of a polynomial is the same as that of the term with the greatest degree. Example 3 Degree of a Polynomial Find the degree of each polynomial. a. x2 ϩ 3x Ϫ 2 b. a2 ϩ ab2 ϩ b4 term degree term degree x 2 2 a 2 2 3x 1 ab 2 1 ϩ 2 or 3 20 b4 4 The greatest degree is 2. So The greatest degree is 4. So the the degree of x2 ϩ 3x Ϫ 2 is 2. degree of a2 ϩ ab2 ϩ b4 is 4. Ecologist Example 4 Degree of a Real-World Polynomial An ecologist studies the relationships between ECOLOGY In the early 1900s, the deer population of the Kaibab organisms and their Plateau in Arizona was affected by hunters and by the food supply. environment. The population from 1905 to 1930 can be approximated by the polynomial Ϫ0.13x5 ϩ 3.13x4 ϩ 4000, where x is the number of years since Online Research 1900. Find the degree of the polynomial. For more information Ϫ 5 ϩ 4 ϩ about a career as an 0.13x 3.13x 4000 Ά Ά Ά ecologist, visit: degree 5 degree 4 degree 0 www.pre-alg.com/ careers So, Ϫ0.13x5 ϩ 3.13x4 ϩ 4000 has degree 5. Concept Check Find the degree of the polynomial at the beginning of the lesson. Concept Check 1. Explain how to find the degree of a monomial and the degree of a polynomial. 2. OPEN ENDED Write three binomial expressions. Explain why they are binomials. 670 Chapter 13 Polynomials and Nonlinear Functions 668-673 C13L1-865108 11/6/03 5:42 PM Page 671 3. FIND THE ERROR Carlos and Tanisha are finding the degree of 5x ϩ y2. Carlos Tanisha 5x has degree 1. 5x has degree 1. y2 has degree 2. y 2 has degree 2. 5x + y2 has degree 1 + 2 or 3. 5x+y2 has degree 2. Who is correct? Explain your reasoning. Guided Practice Determine whether each expression is a polynomial. If it is, classify it as a monomial, binomial, or trinomial. d 1 4. Ϫ7 5. ᎏᎏ 6. ᎏᎏ Ϫ x 2 x 7. a5 ϩ a3 8. y2 Ϫ 4 9. x2 ϩ xy2 Ϫ y2 Find the degree of each polynomial. 10. 4b2 11. 121 12. 8x3y2 13. 3x ϩ 5 14. r3 ϩ 7r 15. d2 ϩ c4 Application GEOMETRY For Exercises 16 and 17, x refer to the square at the right with a side length of x units. 16. Write a polynomial expression for the x area of the small blue rectangle.