What Did You Learn? Key Terms. Key Concepts. 68 Chapter P Prerequisites

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1 7_0P0R.qp /7/06 9:9 AM Page Chapter P Prerequisites What Did You Learn? Key Terms real numbers, p. rational and irrational numbers, p. absolute value, p. variables, p. 6 algebraic epressions, p. 6 Basic Rules of Algebra, p. 7 Zero-Factor Property, p. 8 Key Concepts eponential form, p. scientific notation, p. 4 square root, cube root, p. conjugate, p. 8 polynomial in, p. 4 FOIL Method, p. completely factored, p. 7 domain, p. 7 rational epression, p. 7 rectangular coordinate system, p. 48 Distance Formula, p. 0 Midpoint Formula, p. standard form of the equation of a circle, p. P. Using the Basic Rules of Algebra The properties of real numbers are also true for variables and epressions and are called the Basic Rules of Algebra. P. Using the properties of eponents The properties of eponents can be used when the eponent is an integer or a rational number. P. Simplifying radicals. Let a and b be real numbers and let n be a positive integer. If a b n, then b is an nth root of a.. An epression involving radicals is in simplest form when all possible factors have been removed from the radical, all fractions have radical-free denominators, and the inde of the radical is reduced.. If a is a real number and n and m are positive integers such that the principal nth root of a eists, then a m n a n m a n m and a m n a m n a n m. P. Operations with polynomials. Add or subtract like terms (terms having the eact same variables to the eact same powers) by adding their coefficients.. To find the product of two polynomials, use the left and right Distributive Properties. P. Factoring polynomials. Writing a polynomial as a product is called factoring. If a polynomial cannot be factored using integer coefficients, it is prime or irreducible over the integers.. Factor a polynomial by removing a common factor, by recognizing special product forms, and/or by grouping. P.4 Operations with rational epressions. To add or subtract rational epressions, first rewrite the epressions with the LCD. Then add or subtract the numerators and place over the LCD.. To multiply rational epressions, multiply the numerators, multiply the denominators, and then simplify.. To divide two rational epressions, invert the divisor and multiply. P. Plotting points in the Cartesian plane Each point in the plane corresponds to an ordered pair, y of real numbers and y, called the coordinates of the point. The -coordinate represents the directed distance from the y-ais to the point, and the y-coordinate represents the directed distance from the -ais to the point. P. Using the Distance and Midpoint Formulas. The distance d between points, y and, y in the plane is d y y.. The midpoint of the line segment joining the points, y and, y is given by the Midpoint Formula. Midpoint, y y P.6 Using line plots, histograms, bar graphs, and line graphs to represent data. Use a line plot when ordering small sets of numbers by hand.. Use a histogram when organizing large sets of data.. Use a bar graph when the labels of the bars are not necessarily numbers. The bars of a bar graph can be horizontal or vertical. 4. Use a line graph to show trends over periods of time.

2 7_0P0R.qp /7/06 9:9 AM Page 69 Review Eercises 69 Review Eercises P. In Eercises and, determine which numbers are (a) natural numbers, (b) whole numbers, (c) integers, (d) rational numbers, and (e) irrational numbers.., 4, 8 9,, 6, 0.4.,, 0, 0,., 7 In Eercises and 4, use a calculator to find the decimal form of each rational number. If it is a nonterminating decimal, write the repeating pattern. Then plot the numbers on the real number line and place the correct inequality symbol < or > between them.. (a) (b) 4. (a) (b) In Eercises and 6, verbally describe the subset of real numbers represented by the inequality. Then sketch the subset on the real number line. In Eercises 7 and 8, use the interval notation to describe the set > In Eercises 9 and 0, find the distance between a and b. 9. a 74, b a, b 9 In Eercises 4, use absolute value notation to describe the situation.. The distance between and 7 is at least 6.. The distance between and is no more than 0.. The distance between y and 0 is less than. 4. The distance between y and 6 is greater than 8. In Eercises 8, evaluate the epression for each value of. (If not possible, state the reason.) Epression Values. 9 (a) (b) 6. 4 (a) (b) 7. (a) (b) 8. 4 (a) (b) 9 See for worked-out solutions to odd-numbered eercises. In Eercises 9, identify the rule of algebra illustrated by the statement y 4 0., y 4 y 4. t t. 0 a a In Eercises 8, perform the operation(s). (Write fractional answers in simplest form.) P. In Eercises 9, simplify each epression. 9. (a) z (b) a b 4 ab 8y 40 b 0. (a) 0 (b) y 7 b 6. (a) u v (b) u v. (a) y (b) m n 9 mn y y In Eercises 8, write the number in scientific notation..,8,000,000 4.,0, Sales of The Hershey Company in 006: $,00,000,000 (Source: Value Line) 8. Number of meters in one foot: In Eercises 9 44, write the number in decimal notation Distance between the sun and Jupiter: miles 44. Ratio of day to year:.74 0 In Eercises 4 and 46, use the properties of radicals to simplify the epression

3 7_0P0R.qp /7/06 9:9 AM Page Chapter P Prerequisites In Eercises 47, simplify by removing all possible factors from the radical. 47. a y 4 In Eercises 8, simplify the epression y 6 y Strength of a Wooden Beam In Eercises 9 and 60, use the figure, which shows the rectangular cross section of a wooden beam cut from a log of diameter 4 inches. 9. Find the area of the cross section when w inches and h 4 inches. What is the shape of the cross section? Eplain. 60. The rectangular cross section will have a maimum strength when w 8 inches and h 4 8 inches. Find the area of the cross section. h In Eercises 7 78, perform the operations and write the result in standard form y y y a a a y y y y y 78. In Eercises 79 84, find the special product m 4 n m 4 n 84. y 6 y 6 8. Geometry Use the area model to write two different epressions for the area. Then equate the two epressions and name the algebraic property that is illustrated. In Eercises 6 and 6, rationalize the denominator of the epression. Then simplify your answer In Eercises 6 and 64, rationalize the numerator of the epression. Then simplify your answer In Eercises 6 68, simplify the epression P. In Eercises 69 and 70, write the polynomial in standard form. Then identify the degree and leading coefficient of the polynomial w 86. Compound Interest After years, an investment of $00 compounded annually at an interest rate r will yield an amount of 00 r. Write this polynomial in standard form. In Eercises 87 9, factor out the common factor b Geometry The surface area of a right circular cylinder is S r rh. (a) Draw a right circular cylinder of radius r and height h. Use the figure to eplain how the surface area formula is obtained. (b) Factor the epression for surface area. 94. Business The revenue for selling units of a product at a price of p dollars per unit is R p. For a flat panel television the revenue is R Factor the epression and determine an epression that gives the price in terms of.

4 7_0P0R.qp /7/06 9:9 AM Page 7 Review Eercises 7 In Eercises 9 0, factor the epression In Eercises 0 06, factor by grouping P.4 In Eercises 07 0, find the domain of the epression , > In Eercises 4, write the rational epression in simplest form In Eercises, perform the operations and simplify your answer In Eercises and 4, simplify the comple fraction. y y 6y y P. In Eercises 8, plot the point in the Cartesian plane and determine the quadrant in which it is located.. 8, 6. 4, 9 7., , 0. In Eercises 9 and 0, determine the quadrant(s) in which, y is located so that the conditions are satisfied. 9. > 0 and y 0. y 4 Revenue In Eercises and, use the table, which shows the operating revenues (in millions of dollars) for the motion picture industry for the years 998 to 00. (Source: U.S. Census Bureau) Year. Sketch a scatter plot of the data.. What statement can be made about the operating revenue for the motion picture industry? In Eercises and 4, plot the points and find the distance between the points.., 8,, 4..6, 0, 0, 8. In Eercises and 6, plot the points and find the midpoint of the line segment joining the points..,, 4, , 7.4, 0.6, 4. In Eercises 7 and 8, write the standard form of the equation of the specified circle. 7. Center:, ; solution point:, 8. Endpoints of a diameter: 4, 6, 0, In Eercises 9 and 40, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in the new position. 9. Original coordinates of vertices: 4, 8, 6, 8, 4,, 6, Shift: three units downward, two units to the left 40. Original coordinates of vertices: 0,,,, 0,,, Operating revenue (in millions of dollars) ,00 999, ,040 00, , ,096 Shift: five units upward, four units to the right

5 7_0P0R.qp /7/06 9:9 AM Page 7 7 Chapter P Prerequisites P.6 4. Consumer Awareness Use a line plot to organize the following sample of prices (in dollars) of running shoes. Which price occurred with the greatest frequency? 00, 6, 67, 88, 69, 60, 00, 00, 88, 79, 99, 7, 6, 89, 68, 74, 00, 66, 8, 9, 7, 69, 8, 9, 7 4. Veterans The list shows the numbers of Gulf War veterans (in thousands) in the 0 states and District of Columbia from 990 to 004. Use a frequency distribution and a histogram to organize the data. (Source: Department of Veterans Affairs) AK 8 AL 8 AR 46 AZ 9 CA 6 CO 89 CT 8 DC 6 DE FL 77 GA 79 HI 0 IA 7 ID 8 IL IN 84 KS 4 KY 6 LA 7 MA 4 MD 9 ME 0 MI 6 MN MO 8 MS 49 MT 6 NC 4 ND 0 NE 8 NH 8 NJ 6 NM NV 4 NY 7 OH OK 6 OR PA 4 RI SC 86 SD TN 97 TX 4 UT 9 VA 96 VT 7 WA WI 6 WV 7 WY 4. Meteorology The normal daily maimum and minimum temperatures (in F) for each month for the city of Chicago are shown in the table. Construct a double bar graph for the data. (Source: National Climatic Data Center) Month Ma. Min. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Law Enforcement The table shows the numbers of people indicted for public corruption in the United States from 99 to 00. Construct a line graph for the data and state what information the graph reveals. (Source: U.S. Department of Justice) Year TABLE FOR 44 Number of indictments Basketball The list shows the average numbers of points per game for the top 0 NBA players for the regular NBA season. Organize the data in an appropriate display. Eplain your choice of graph. (Source: National Basketball Association) 0.7, 7.6, 7., 6., 6.0,.7,., 4.6, 4., 4., 4.,.9,.0,.9,.,.,.,.0,.7, Salaries The table shows the average salaries (in thousands of dollars) for professors, associate professors, assistant professors, and instructors at public institutions of higher education from 00 to 00. Organize the data in an appropriate display. Eplain your choice of graph. (Source: American Association of University Professors) Rank Professor Associate Professor Assistant Professor.. 4. Instructor Synthesis True or False? In Eercises 47 and 48, determine whether the statement is true or false. Justify your answer. 47. for all values of. 48. A binomial sum squared is equal to the sum of the terms squared. Error Analysis In Eercises 49 and 0, describe the error Writing Eplain why u u u.

6 7_0P0R.qp /7/06 9:40 AM Page 7 Chapter Test 7 P Chapter Test See for worked-out solutions to odd-numbered eercises. Take this test as you would take a test in class. After you are finished, check your work against the answers in the back of the book.. Use < or > to show the relationship between 0 and. Find the distance between the real numbers 7 and 9.. Identify the rule of algebra illustrated by 0. In Eercises 4 and, evaluate each epression without using a calculator. 4. (a) (b) (c) (d) (a) 8 (b) (c) (d) In Eercises 6 and 7, simplify each epression. 6. (a) z z (b) u 4 u (c) 7. (a) 9z 8z z (b) 6y 0 y (c) 8. Write the polynomial 4 in standard form. Identify the degree and leading coefficient. In Eercises 9, perform the operations and simplify y 4 6 v In Eercises, find the special product y z y z In Eercises 6 8, factor the epression completely Rationalize each denominator: (a) (b) and (c) 6,,. 0. Write an epression for the area of the shaded region in the figure at the right and simplify the result.. Plot the points, and 6, 0. Find the coordinates of the midpoint of the line segment joining the points and the distance between the points.. The numbers (in millions) of votes cast for the Democratic candidates for president in 980, 984, 988, 99, 996, 000, and 004 were., 7., 4.7, 44.9, 47.4,.0, and 8.9, respectively. Construct a bar graph for the data. (Source: Office of the Clerk, U.S. House of Representatives) Figure for 0

7 7_0P0R.qp /7/06 9:40 AM Page Chapter P Prerequisites Proofs in Mathematics What does the word proof mean to you? In mathematics, the word proof is used to mean simply a valid argument. When you are proving a statement or theorem, you must use facts, definitions, and accepted properties in a logical order. You can also use previously proved theorems in your proof. For instance, the Distance Formula is used in the proof of the Midpoint Formula below. There are several different proof methods, which you will see in later chapters. The Midpoint Formula (p. ) The midpoint of the line segment joining the points, y and, y is given by the Midpoint Formula Midpoint, y y. Proof Using the figure, you must show that d d and d d d. By the Distance Formula, you obtain d y (, y ) d d d y y ( +, y y + y (, y ) ( The Cartesian Plane The Cartesian plane was named after the French mathematician René Descartes (96 60). While Descartes was lying in bed, he noticed a fly buzzing around on the square ceiling tiles. He discovered that the position of the fly could be described by which ceiling tile the fly landed on. This led to the development of the Cartesian plane. Descartes felt that a coordinate plane could be used to facilitate description of the positions of objects. y y d y y y y y d y y. So, it follows that d d and d d d.