Exponents, Polynomials, and Polynomial Functions

Transcription

1 CHAPTER Exponents, Polynomials, and Polynomial Functions. Exponents and Scientific Notation. More Work with Exponents and Scientific Notation. Polynomials and Polynomial Functions. Multiplying Polynomials. The Greatest Common Factor and Factoring by Grouping.6 Factoring Trinomials.7 Factoring by Special Products Integrated Review Operations on Polynomials and Factoring Strategies.8 Solving Equations by Factoring and Problem Solving Linear equations are important for solving problems. They are not sufficient, however, to solve all problems. Many real-world phenomena are modeled by polynomials. We begin this chapter by reviewing exponents. We will then study operations on polynomials and how polynomials can be used in problem solving. A hybrid vehicle is one with a gasoline engine and an electric motor, each of which is used to propel the vehicle. With the arrival of electric-drive cars, it is very difficult to predict future sales of hybrids based on past sales, but we will attempt to do so. In Section.8, Exercises 0 and 0, we attempt to predict sales of hybrids in future years. Honda 9% Ford % Hybrid Sales for a Month in 00 GM % Nissan % Toyota 7%

2 Section. Exponents and Scientific Notation. Exponents and Scientific Notation S Use the Product Rule for Exponents. Evaluate Expressions Raised to the 0 Power. Use the Quotient Rule for Exponents. Evaluate Expressions Raised to the Negative nth Power. Convert Between Scientific Notation and Standard Notation. Using the Product Rule Recall that exponents may be used to write repeated factors in a more compact form. As we have seen in the previous chapters, exponents can be used when the repeated factor is a number or a variable. For example, base base y 6 exponent means exponent means # # factors; each factor is y # y # y # y # y # y 6 factors; each factor is y Expressions such as and y 6 that contain exponents are called exponential expressions. Exponential expressions can be multiplied, divided, added, subtracted, and themselves raised to powers. In this section, we review operations on exponential expressions. We review multiplication first. To multiply x by x, use the definition of an exponent. x # x = x # xx # x # x x is a factor times = x Notice that the result is exactly the same if we add the exponents. x # x = x + = x This suggests the following. Product Rule for Exponents If m and n are positive integers and a is a real number, then a m # a n = a m +n In other words, the product of exponential expressions with a common base is the common base raised to a power equal to the sum of the exponents of the factors. EXAMPLE a. # a. # = + = 7 b. x 7 x = x 7 + = x 0 c. y # y # y = y # y # y = y # y = y 7 Use the product rule to simplify. b. x 7 x c. y # y # y Use the product rule to simplify. a. # b. x # x c. y # y # y

3 6 CHAPTER Exponents, Polynomials, and Polynomial Functions EXAMPLE Use the product rule to simplify. a. x 6 x b. -.x p xp 0 Here, we use properties of multiplication to group together like bases. a. x 6 x = x 6 x = x 7 b. -.x p xp 0 = -.x x p p 0 = -9.6x p Use the product rule to simplify. a. z 7z b. -.t q tq Evaluating Expressions Raised to the 0 Power The definition of a n does not include the possibility that n might be 0. But if it did, then, by the product rule, a 0 # a n = a 0 +n = a n = # a n ()* ()* From this, we reasonably define that a 0 = as long as a does not equal 0. Zero Exponent If a does not equal 0, then a 0 =. EXAMPLE Evaluate the following. a. 7 0 b c. x + 0 d. x 0 a. 7 0 = b. Without parentheses, only 7 is raised to the 0 power = -7 0 = - = - c. x + 0 = d. x 0 = = Evaluate the following. a. 0 b. - 0 c. x d. x 0 Using the Quotient Rule To find quotients of exponential expressions, we again begin with the definition of a n to simplify x9. For example, x x 9 x = x # x # x # x # x # x # x # x # x x # x = x 7 (Assume for the next two sections that denominators containing variables are not 0.) Notice that the result is exactly the same if we subtract the exponents. x 9 x = x9 - = x 7

4 Section. Exponents and Scientific Notation 7 This suggests the following. Quotient Rule for Exponents If a is a nonzero real number and n and m are integers, then a m -n n = am a In other words, the quotient of exponential expressions with a common base is the common base raised to a power equal to the difference of the exponents. EXAMPLE a. x7 x b. 8 a. x7 x = x7 - = x b. 8 = 8 - = 6 c. 0x6 x d. y0 z 7 y 8 z 7 a. z8 z Use the quotient rule to simplify. 0x6 c. x d. y0 z 7 y 8 z 7 = x 6 - = x, or x = 6 7 y0-8 # z 7-7 = 6 7 y z 0 = 6 7 y, or 6y 7 Use the quotient rule to simplify. b. 9 x7 c. x d. a b 6 8a 7 b 6 Evaluating Exponents Raised to the Negative nth Power When the exponent of the denominator is larger than the exponent of the numerator, applying the quotient rule yields a negative exponent. For example, x x = x - = x - Using the definition of a n, though, gives us x x = x # x # x x # x # x # x # x = x From this, we reasonably define x - = x or, in general, a-n = a n. Negative Exponents If a is a real number other than 0 and n is a positive integer, then a -n = a n

5 8 CHAPTER Exponents, Polynomials, and Polynomial Functions EXAMPLE Simplify and write with positive exponents only. a. - b. - - c. x - d. x - e. m f. m 6 g h. t - a. - = = b. - - = - = 6 x - = c. x - = # x = x Without parentheses, only x is raised to the - power. x d. x - = x = With parentheses, both and x are raised to the - power. x e. m m = m - = m -0 = m 0 f. 6 = -6 = - = = 7 g = + = + 9 = = 8 h. t - = t =, t = # t = t Helpful Hint Study Example c. Make sure you understand that Simplify and write with positive exponents only. a. 6 - b c. x - d. y - e. f. g h. z -8 k k Helpful Hint Notice that when a factor containing an exponent is moved from the numerator to the denominator or from the denominator to the numerator, the sign of its exponent changes. x - = x, - = = y - = y, - = = 8 EXAMPLE 6 a. x-9 x p b. p Simplify and write with positive exponents only. - c. - a. x-9 x = x -9 - = x - = x b. p p - = # p -- = p 7 - d. x-7 y 0xy - e. x- x x 6

6 Section. Exponents and Scientific Notation 9 c. - - = - -- = - = = d. x-7 y 0xy - = x-7 - # y -- = x-8 y 7 = y7 x 8 e. Simplify the numerator first. x - x x 6 = x- + x 6 = x- x 6 = x - -6 = x -7 = x 7 6 Simplify and write with positive exponents only. a. z-8 z 7t b. t - c. - - d. a - b a b - e. x- x 6 x CONCEPT CHECK Find and correct the error in the following: y -6 y - = y-6 - = y -8 = y 8 EXAMPLE 7 is not 0. Simplify. Assume that a and t are nonzero integers and that x a. x a # x b. xt- x t- a. x a # x = x a + Use the product rule. b. xt- x t- = xt- -t- Use the quotient rule. = x t--t+ = x t+ 7 Simplify. Assume that a and t are nonzero integers and that x is not 0. a. x a # x b. xt- x t- Converting Between Scientific Notation and Standard Notation Very large and very small numbers occur frequently in nature. For example, the distance between the earth and the sun is approximately 0,000,000 kilometers. A helium atom has a diameter of centimeter. It can be tedious to write these very large and very small numbers in standard notation like this. Scientific notation is a convenient shorthand notation for writing very large and very small numbers. Helium atom Answer to Concept Check: y -6 y - = y = y - = y centimeter 0,000,000 km

7 60 CHAPTER Exponents, Polynomials, and Polynomial Functions Scientific Notation A positive number is written in scientific notation if it is written as the product of a number a, where a 6 0, and an integer power r of 0: a * 0 r The following are examples of numbers written in scientific notation. diameter of helium atom:. * 0-8 cm approximate distance between Earth and the sun:. * 0 8 km Writing a Number in Scientific Notation Step. Step. Step. Move the decimal point in the original number until the new number has a value between and 0. Count the number of decimal places the decimal point was moved in Step. If the original number is 0 or greater, the count is positive. If the original number is less than, the count is negative. Write the product of the new number in Step by 0 raised to an exponent equal to the count found in Step. EXAMPLE 8 Write each number in scientific notation. a. 70,000 b a. Step. Move the decimal point until the number is between and 0. 70,000. Step. The decimal point is moved places and the original number is 0 or greater, so the count is positive. Step. 70,000 = 7. * 0. b. Step. Move the decimal point until the number is between and Step. The decimal point is moved 6 places and the original number is less then, so the count is -6. Step =.0 * Write each number in scientific notation. a. 6,000 b To write a scientific notation number in standard form, we reverse the preceding steps. Writing a Scientific Notation Number in Standard Notation Move the decimal point in the number the same number of places as the exponent on 0. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left.

8 Section. Exponents and Scientific Notation 6 Answers to Concept Check: a, c, d EXAMPLE 9 Write each number in standard notation. a. 7.7 * 0 8 b..0 * 0 - a. 7.7 * 0 8 = 770,000,000 Since the exponent is positive, move the decimal point 8 places to the right. Add zeros as needed. b..0 * 0 - = Since the exponent is negative, move the decimal point places to the left. Add zeros as needed. 9 Write each number in standard notation. a. 6. * 0 b..09 * 0 - CONCEPT CHECK Which of the following numbers have values that are less than? a.. * 0 - b.. * 0 c. -. * 0 d. -. * 0 - Scientific Calculator Explorations Multiply,000,000 by 700,000 on your calculator. The display should read. or. E, which is the product written in scientific notation. Both these notations mean. * 0. To enter a number written in scientific notation on a calculator, find the key marked EE. (On some calculators, this key may be marked EXP.) To enter 7.6 * 0, press the keys 7.6 EE The display will read 7.6 or 7.6 E. Use your calculator to perform each operation indicated.. Multiply * 0 and * 0.. Divide 6 * 0 by * Multiply. * 0 and 7. * 0.. Divide.8 * 0 by * 0 7. Vocabulary, Readiness & Video Check State the base of the exponent in each expression.. 9x. yz y # Martin-Gay Interactive Videos See Video. Watch the section lecture video and answer the following questions. 7. Why are we reminded that multiplication is commutative and associative during the simplifying of Example? 8. Example does not contain parentheses yet a discussion of parentheses is an important part of evaluating. Explain. 9. When applying the quotient rule in Example, how do you know which exponents to subtract? 0. What important reminder is made at the beginning of Example 6?. From Examples 0 and, explain how the direction of the movement of the decimal point in Step might suggest another way to determine the sign of the exponent on 0.

9 6 CHAPTER Exponents, Polynomials, and Polynomial Functions. Exercise Set Write each expression with positive exponents. See Examples and 6.. x - y -. 7xy -. a b - c -. a - b c - 6. y - x - 6. x - 7 z - Use the product rule to simplify each expression. See Examples and. 7. # 8. # 9. x # x. m # m 7 # m 6 0. a # a 9. n # n 0 # n. xy-x. -7xy7y. -x p y x 6. -6a b -ab Evaluate each expression. See Example x x x 0. -x 0. x x 0 + Use the quotient rule to simplify. See Example.. a a 6. x9 x z z x 8x 9. x 9 y 6 0. a b x 8 y 6 a 9 b. x y 7. a0 b 9xy 0ab. -6a b 7 c 0 6ab c. 9a bc -7abc 8 Simplify and write using positive exponents only. See Examples and z x z. a -. 0b x 7 x - 7 y -. p - q - x - 6. z - x z 0 8r s r - 8. s - x - 9 x 0. y - 7 y x - y 8 a - 6 b. 8ab - 6 8ab - a - b 6 x 8 x 0x z z z - MIXED Simplify and write using positive exponents only. See Examples through x # 0x y # -9y 7. x 7 # x 8 # x 8. y 6 # y # y 9 9. x # x z # 0z 7 6. x 0 + x 0 6. y 0 - y 0 6. z z 6. x x t x y - y y - 6 y x - 7. x - 7. r r - 7. x - x 7. x - 7 y - x y 76. a - b 7 a - b x yx -xy a bb -ab # x # y a - 9 b a b - x - yz x - 7 yz - xyz xyz Simplify. Assume that variables in the exponents represent nonzero integers and that x, y, and z are not 0. See Example x # x 7a 90. x a # x 7 9. x t - 9. y p # y 9p z 6x x t 9. yp- y p 9. x 9y # x - 7y z y z x t # x t - # x t 98. zx z x - 7 z x Write each number in scientific notation. See Example ,0, , , 0. 6,800, ,700,000,000 Write each number in scientific notation. 09. The approximate distance between Jupiter and the sun is 778,00,000 kilometers. (Source: National Space Data Center) y 6

10 Section. Exponents and Scientific Notation 6 0. For the 00 0 National Football League season, the payroll of the Super Bowl champion Green Bay Packers was approximately $7,,00. (Source: USA Today). The gross domestic product (GDP) of a region is the value of all final goods and services produced within a given year. The GDP of the European Union for 00 was estimated to be equivalent to U.S. $6,8,000,000,000. (Source: Wikipedia). By the end of 00, Nintendo had sold,0,000 Wii units. (Source: Nintendo). Lake Mead, created from the Colorado River by the Hoover Dam, has a capacity of,000,000,000 cubic feet of water. (Source: U.S. Bureau of Reclamation) 0. Hagerman Fossil Beds National Monument in Idaho contains 6.68 * 0 - square kilometers of privately owned land.. In 00, the Coca-Cola Company sold. * 0 0 unit cases of beverages worldwide. (Source: the Coca-Cola Company). In 00, the revenue for McDonald s was +.7 * 0 0. (Source: McDonald s Corporation). The temperature of the core of the sun is about 7,000,000 F. Core. A pulsar is a rotating neutron star that gives off sharp, regular pulses of radio waves. For one particular pulsar, the rate of pulses is every 0.00 second. 6. To convert from cubic inches to cubic meters, multiply by Write each number in standard notation, without exponents. See Example * * * * * * * * * * 0 8 Write each number in standard notation. 7. The estimated world population in C.E. was.0 * 0 8. (Source: World Almanac and Book of Facts) 8. There are.99 * 0 6 miles of highways, roads, and streets in the United States. (Source: Bureau of Transportation Statistics) 9. Chiricahua National Monument in Arizona contains 9. * 0 - square kilometers of privately owned land. REVIEW AND PREVIEW Evaluate. See Sections. and... #. #. a b CONCEPT EXTENSIONS 9. Explain how to convert a number from standard notation to scientific notation. 0. Explain how to convert a number from scientific notation to standard notation.. Explain why - 0 simplifies to but - 0 simplifies to -.. Explain why both x 0 - y 0 and x - y 0 simplify to.. Simplify where possible. a. x a # x a b. x a + x a c. xa x b d. x a # x b e. x a + x b. Which numbers are equal to 6,000? Of these, which is written in scientific notation? a. 6 * 0 b. 60 * 0 c. 0.6 * 0 d..6 * 0 Without calculating, determine which number is larger.. 7 or or or or -9