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  ### Pre-calculus
  
  1. 1. 6 CHAPTER R BASIC ALGEBRAIC OPERATIONS Z BASIC PROPERTIES OF THE SET OF REAL NUMBERS Let R be the set of real numbers, and let x, y, and z be arbitrary elements of R. Addition Properties Closure: x ϩ y is a unique element in R. Associative: (x ϩ y) ϩ z ϭ x ϩ ( y ϩ z) Commutative: xϩyϭyϩx Identity: 0 is the additive identity; that is, 0 ϩ x ϭ x ϩ 0 ϭ x for all x in R, and 0 is the only element in R with this property. Inverse: For each x in R, Ϫx is its unique additive inverse; that is, x ϩ (Ϫx) ϭ (Ϫx) ϩ x ϭ 0, and Ϫx is the only element in R relative to x with this property. Multiplication Properties Closure: xy is a unique element in R. Associative: (xy)z ϭ x( yz) Commutative: xy ϭ yx Identity: 1 is the multiplicative identity; that is, for all x in R, (1)x ϭ x(1) ϭ x, and 1 is the only element in R with this property. Inverse: For each x in R, x 0, xϪ1 is its unique multiplicative inverse; that is, xxϪ1 ϭ xϪ1x ϭ 1, and xϪ1 is the only element in R relative to x with this property. Combined Property Distributive: x( y ϩ z) ϭ xy ϩ xz (x ϩ y)z ϭ xz ϩ yz EXAMPLE 2 Using Real Number Properties Which real number property justiﬁes the indicated statement? (A) (7x)y ϭ 7(xy) (B) a(b ϩ c) ϭ (b ϩ c)a (C) (2x ϩ 3y) ϩ 5y ϭ 2x ϩ (3y ϩ 5y) (D) (x ϩ y)(a ϩ b) ϭ (x ϩ y)a ϩ (x ϩ y)b (E) If a ϩ b ϭ 0, then b ϭ Ϫa. SOLUTIONS (A) Associative (ؒ) (B) Commutative (ؒ) (C) Associative (ϩ) (D) Distributive (E) Inverse (ϩ)
  2. 2. SECTION R–1 Algebra and Real Numbers 7MATCHED PROBLEM 2 Which real number property justiﬁes the indicated statement? (A) 4 ϩ (2 ϩ x) ϭ (4 ϩ 2) ϩ x (B) (a ϩ b) ϩ c ϭ c ϩ (a ϩ b) (C) 3x ϩ 7x ϭ (3 ϩ 7)x (D) (2x ϩ 3y) ϩ 0 ϭ 2x ϩ 3y (E) If ab ϭ 1, then b ϭ 1րa. Z Further Operations and Properties Subtraction of real numbers can be deﬁned in terms of addition and the additive inverse. If a and b are real numbers, then a Ϫ b is deﬁned to be a ϩ (Ϫb). Similarly, division can be deﬁned in terms of multiplication and the multiplicative inverse. If a and b are real num- bers and b 0, then a Ϭ b (also denoted a͞b) is deﬁned to be a ؒ bϪ1. Z DEFINITION 2 Subtraction and Division of Real Numbers For all real numbers a and b: Subtraction: a Ϫ b ϭ a ϩ (Ϫb) 5 ؊ 3 ‫2 ؍ )3؊( ؉ 5 ؍‬ Division: a Ϭ b ϭ a ؒ bϪ1 b 0 3 ، 2 ‫ؒ 3 ؍ 1؊2 ؒ 3 ؍‬ 1 2 ‫5.1 ؍‬ It is important to remember that Division by 0 is never allowed.ZZZ EXPLORE-DISCUSS 1 (A) Give an example that shows that subtraction of real numbers is not commutative. (B) Give an example that shows that division of real numbers is not commutative. The basic properties of the set of real numbers, together with the deﬁnitions of sub- traction and division, lead to the following properties of negatives and zero. Z THEOREM 1 Properties of Negatives For all real numbers a and b: 1. Ϫ(Ϫa) ϭ a 2. (Ϫa)b ϭ Ϫ(ab) ϭ a(Ϫb) ϭ Ϫab 3. (Ϫa)(Ϫb) ϭ ab 4. (Ϫ1)a ϭ Ϫa Ϫa a a 5. ϭϪ ϭ b 0 b b Ϫb Ϫa Ϫa a a 6. ϭϪ ϭϪ ϭ b 0 Ϫb b Ϫb b
  3. 3. 8 CHAPTER R BASIC ALGEBRAIC OPERATIONS Z THEOREM 2 Zero Properties For all real numbers a and b: 1. a ؒ 0 ϭ 0 ؒ a ϭ 0 2. ab ϭ 0 if and only if* a ϭ 0 or b ϭ 0 or both Note that if b 0, then 0 ϭ 0 ؒ bϪ1 ϭ 0 by Theorem 2. In particular, 0 ϭ 0; but the expres- b 3 sions 3 and 0 are undeﬁned. 0 0 EXAMPLE 3 Using Negative and Zero Properties Which real number property or deﬁnition justiﬁes each statement? (A) 3 Ϫ (Ϫ2) ϭ 3 ϩ [Ϫ(Ϫ2)] ϭ 5 (B) Ϫ(Ϫ2) ϭ 2 Ϫ3 3 (C) Ϫ ϭ 2 2 5 5 (D) ϭϪ Ϫ2 2 (E) If (x Ϫ 3)(x ϩ 5) ϭ 0, then either x Ϫ 3 ϭ 0 or x ϩ 5 ϭ 0. SOLUTIONS (A) Subtraction (Deﬁnition 1 and Theorem 1, part 1) (B) Negatives (Theorem 1, part 1) (C) Negatives (Theorem 1, part 6) (D) Negatives (Theorem 1, part 5) (E) Zero (Theorem 2, part 2) MATCHED PROBLEM 3 Which real number property or deﬁnition justiﬁes each statement? 3 1 (A) ϭ 3a b (B) (Ϫ5)(2) ϭ Ϫ(5 ؒ 2) (C) (Ϫ1)3 ϭ Ϫ3 5 5 Ϫ7 7 (D) ϭϪ (E) If x ϩ 5 ϭ 0, then (x Ϫ 3)(x ϩ 5) ϭ 0. 9 9ZZZ EXPLORE-DISCUSS 2 A set of numbers is closed under an operation if performing the operation on num- bers of the set always produces another number in the set. For example, the set of odd integers is closed under multiplication, but is not closed under addition. (A) Give an example that shows that the set of irrational numbers is not closed under addition. (B) Explain why the set of irrational numbers is closed under taking multiplica- tive inverses. *Given statements P and Q, “P if and only if Q” stands for both “if P, then Q” and “if Q, then P.”
  4. 4. SECTION R–1 Algebra and Real Numbers 9 If a and b are real numbers, b 0, the quotient a Ϭ b, when written in the form a͞b, is called a fraction. The number a is the numerator, and b is the denominator. It can be shown that fractions satisfy the following properties. (Note that some of these properties, under the restriction that numerators and denominators are integers, were used earlier to deﬁne arithmetic operations on the rationals.) Z THEOREM 3 Fraction Properties For all real numbers a, b, c, d, and k (division by 0 excluded): a c 1. ϭ if and only if ad ϭ bc b d 4 6 ‫؍‬ since 4ؒ9‫6ؒ6؍‬ 6 9 ka a a c ac a c a d 2. ϭ 3. ؒ ϭ 4. Ϭ ϭ ؒ kb b b d bd b d b c 7ؒ3 3 3 7 3ؒ7 21 2 5 2 7 14 ‫؍‬ ؒ ‫؍‬ ‫؍‬ ، ‫؍ ؒ ؍‬ 7ؒ5 5 5 8 5ؒ8 40 3 7 3 5 15 a c aϩc a c aϪc a c ad ϩ bc 5. ϩ ϭ 6. Ϫ ϭ 7. ϩ ϭ b b b b b b b d bd 3 4 3؉4 7 7 2 7؊2 5 2 1 2ؒ5؉3ؒ1 13 ؉ ‫؍‬ ‫؍‬ ؊ ‫؍‬ ‫؍‬ ؉ ‫؍‬ ‫؍‬ 6 6 6 6 8 8 8 8 3 5 3ؒ5 15 ANSWERS TO MATCHED PROBLEMS 1. (A) Ϫ29ր6 (B) Ϫ17 ր8 (C) 9 ր8 (D) 25ր6 2. (A) Associative (ϩ) (B) Commutative (ϩ) (C) Distributive (D) Identity (ϩ) (E) Inverse (ؒ) 3. (A) Division (Deﬁnition 1) (B) Negatives (Theorem 1, part 2) (C) Negatives (Theorem 1, part 4) (D) Negatives (Theorem 1, part 5) (E) Zero (Theorem 2, part 1) R-1 ExercisesIn Problems 1–16, perform the indicated operations, if deﬁned. If 11 1 2 7 7. Ϭ 8. Ϭthe result is not an integer, express it in the form a/b, where a and 5 3 9 5b are integers. 9. 100 Ϭ 0 10. 0 Ϭ 0 1 1 1 1 1. ϩ 2. ϩ 3 5 4 6 3 5 2 7 11. aϪ b aϪ b 12. Ϭ a3 Ϫ b 5 3 7 2 3 4 8 4 3. Ϫ 4. Ϫ 17 2 2 5 4 3 9 5 13. ؒ 14. aϪ b aϪ b 8 7 3 6 2 4 1 3 5. ؒ 6. aϪ bؒ 3 Ϫ1 3 7 10 8 15. a b ϩ 2Ϫ1 16. Ϫ(4Ϫ1 ϩ 3) 8
  5. 5. 10 CHAPTER R BASIC ALGEBRAIC OPERATIONSIn Problems 17–28, each statement illustrates the use of one of the 39. Indicate true (T) or false (F), and for each false statement ﬁndfollowing properties or deﬁnitions. Indicate which one. real number replacements for a and b that will provide a counterexample. For all real numbers a and b:Commutative (ϩ) Inverse (ϩ) (A) a ϩ b ϭ b ϩ aCommutative (ؒ) Inverse (ؒ) (B) a Ϫ b ϭ b Ϫ aAssociative (ϩ) Subtraction (C) ab ϭ baAssociative (ؒ) Division (D) a Ϭ b ϭ b Ϭ aDistributive Negatives (Theorem 1)Identity (ϩ) Zero (Theorem 2) 40. Indicate true (T) or false (F), and for each false statement ﬁndIdentity (ؒ) real number replacements for a, b, and c that will provide a counterexample. For all real numbers a, b, and c:17. x ϩ ym ϭ x ϩ my 18. 7(3m) ϭ (7 ؒ 3)m (A) (a ϩ b) ϩ c ϭ a ϩ (b ϩ c) u u19. 7u ϩ 9u ϭ (7 ϩ 9)u 20. Ϫ ϭ (B) (a Ϫ b) Ϫ c ϭ a Ϫ (b Ϫ c) Ϫv v (C) a(bc) ϭ (ab)c21. (Ϫ2)(Ϫ2) ϭ 1 1 22. 8 Ϫ 12 ϭ 8 ϩ (Ϫ12) (D) (a Ϭ b) Ϭ c ϭ a Ϭ (b Ϭ c)23. w ϩ (Ϫw) ϭ 0 24. 5 Ϭ (Ϫ6) ϭ 5(Ϫ6) 125. 3(xy ϩ z) ϩ 0 ϭ 3(xy ϩ z) In Problems 41–48, indicate true (T) or false (F), and for each false statement give a speciﬁc counterexample.26. ab(c ϩ d ) ϭ abc ϩ abd Ϫx x 41. The difference of any two natural numbers is a natural number.27. ϭ 28. (x ϩ y) ؒ 0 ϭ 0 Ϫy y 42. The quotient of any two nonzero integers is an integer. 43. The sum of any two rational numbers is a rational number.29. If ab ϭ 0, does either a or b have to be 0? 44. The sum of any two irrational numbers is an irrational number.30. If ab ϭ 1, does either a or b have to be 1? 45. The product of any two irrational numbers is an irrational number.31. Indicate which of the following are true: (A) All natural numbers are integers. 46. The product of any two rational numbers is a rational number. (B) All real numbers are irrational. 47. The multiplicative inverse of any irrational number is an (C) All rational numbers are real numbers. irrational number.32. Indicate which of the following are true: 48. The multiplicative inverse of any nonzero rational number is a (A) All integers are natural numbers. rational number. (B) All rational numbers are real numbers. (C) All natural numbers are rational numbers. 49. If c ϭ 0.151515 . . . , then 100c ϭ 15.1515 . . . and33. Give an example of a rational number that is not an integer. 100c Ϫ c ϭ 15.1515 . . . Ϫ 0.151515 . . .34. Give an example of a real number that is not a rational number. 99c ϭ 15 c ϭ 15 ϭ 33 99 5In Problems 35 and 36, list the subset of S consisting of(A) natural numbers, (B) integers, (C) rational numbers, and Proceeding similarly, convert the repeating decimal 0.090909(D) irrational numbers. . . . into a fraction. (All repeating decimals are rational num- bers, and all rational numbers have repeating decimal repre-35. S ϭ 5Ϫ3, Ϫ2, 0, 1, 13, 9, 11446 3 5 sentations.)36. S ϭ 5Ϫ15, Ϫ1, Ϫ1, 2, 17, 6, 1625ր9, ␲6 2 50. Repeat Problem 49 for 0.181818. . . .In Problems 37 and 38, use a calculator* to express each number indecimal form. Classify each decimal number as terminating,repeating, or nonrepeating and nonterminating. Identify the patternof repeated digits in any repeating decimal numbers. 837. (A) 9 3 (B) 11 (C) 15 (D) 11 838. (A) 13 6 (B) 121 7 (C) 16 29 (D) 111*Later in the book you will encounter optional exercises that require a graphing calculator. If you have such a calculator, you can certainly use it here.Otherwise, any scientiﬁc calculator will be sufﬁcient for the problems in this chapter.
  6. 6. SECTION R–2 Exponents and Radicals 11 R-2 Exponents and Radicals Z Integer Exponents Z Scientiﬁc Notation Z Roots of Real Numbers Z Rational Exponents and Radicals Z Simplifying Radicals The French philosopher/mathematician René Descartes (1596–1650) is generally credited with the introduction of the very useful exponent notation “x n.” This notation as well as other improvements in algebra may be found in his Geometry, published in 1637. If n is a natural number, x n denotes the product of n factors, each equal to x. In this sec- tion, the meaning of x n will be expanded to allow the exponent n to be any rational number. Each of the following expressions will then represent a unique real number: 75 5Ϫ4 3.140 61ր2 14Ϫ5ր3 Z Integer Exponents If a is a real number, then a6 ϭ a ؒ a ؒ a ؒ a ؒ a ؒ a 6 factors of a In the expression a6, 6 is called an exponent and a is called the base. Recall that aϪ1, for a 0, denotes the multiplicative inverse of a (that is, 1 րa). To gen- eralize exponent notation to include negative integer exponents and 0, we deﬁne aϪ6 to be the multiplicative inverse of a6, and we deﬁne a0 to be 1. n Z DEFINITION 1 a , n an Integer and a a Real Number For n a positive integer and a a real number: an ϭ a ؒ a ؒ . . . ؒ a n factors of a 1 aϪn ϭ n (a 0) a a ϭ1 0 (a 0)EXAMPLE 1 Using the Deﬁnition of Integer Exponents Write parts (A) and (B) in decimal form and parts (C) and (D) using positive exponents. Assume all variables represent nonzero real numbers. (A) (u3v2)0 (B) 10Ϫ3 xϪ3 (C) xϪ8 (D) yϪ5
  7. 7. 12 CHAPTER R BASIC ALGEBRAIC OPERATIONS 1 1 SOLUTIONS (A) (u3v2)0 ϭ 1 (B) 10Ϫ3 ϭ ϭ ϭ 0.001 103 1,000 * 1 xϪ3 xϪ3 1 1 y5 y5 (C) xϪ8 ϭ (D) ϭ ؒ Ϫ5 ϭ 3 ؒ ϭ x8 yϪ5 1 y x 1 x3 MATCHED PROBLEM 1 Write parts (A) and (B) in decimal form and parts (C) and (D) using positive exponents. Assume all variables represent nonzero real numbers. 1 uϪ7 (A) (x2)0 (B) 10Ϫ5 (C) Ϫ4 (D) x vϪ3 To calculate with exponents, it is helpful to remember Deﬁnition 1. For example: 23 ؒ 24 ϭ (2 ؒ 2 ؒ 2)(2 ؒ 2 ؒ 2 ؒ 2) ϭ 23ϩ4 ϭ 27 (23)4 ϭ (2 ؒ 2 ؒ 2)4 ϭ (2 ؒ 2 ؒ 2)(2 ؒ 2 ؒ 2)(2 ؒ 2 ؒ 2)(2 ؒ 2 ؒ 2) ϭ 23ؒ4 ϭ 212 These are instances of Properties 1 and 2 of Theorem 1. Z THEOREM 1 Properties of Integer Exponents For n and m integers and a and b real numbers: 1. aman ϭ amϩn a5a ؊7 ‫ ؍‬a5؉(؊7) ‫ ؍‬a؊2 2. (an)m ϭ amn (a3)؊2 ‫ ؍‬a(؊2)3 ‫ ؍‬a؊6 3. (ab)m ϭ ambm (ab)3 ‫ ؍‬a3b3 a m am a 4 a4 4. a b ϭ m b 0 a b ‫4 ؍‬ b ͭ b b b a3 m amϪn a؊2 ‫ ؍‬a3؊(؊2) ‫ ؍‬a5 a 5. n ϭ 1 a 0 a3 1 1 a ‫؍‬ ‫؍‬ anϪm a؊2 a؊2؊3 a؊5 EXAMPLE 2 Using Exponent Properties Simplify using exponent properties, and express answers using positive exponents only.† 6xϪ2 (A) (3a5)(2aϪ3) (B) 8xϪ5 (C) Ϫ4y3 Ϫ (Ϫ4y)3 (D) (2aϪ3b2)Ϫ2 SOLUTIONS (A) (3a5)(2aϪ3) ϭ (3 ؒ 2)(a5aϪ3) ϭ 6a2 6xϪ2 3xϪ2Ϫ(Ϫ5) 3x3 (B) ϭ ϭ 8xϪ5 4 4 *Throughout the book, dashed boxes—called think boxes—are used to represent steps that may be performed mentally. † By “simplify” we mean eliminate common factors from numerators and denominators and reduce to a minimum the number of times a given constant or variable appears in an expression. We ask that answers be expressed using positive exponents only in order to have a deﬁnite form for an answer. Later (in this section and elsewhere) we will encounter situations where we will want negative exponents in a ﬁnal answer.
  8. 8. SECTION R–2 Exponents and Radicals 13 (C) Ϫ4y3 Ϫ (Ϫ4y)3 ϭ Ϫ4y3 Ϫ (Ϫ4)3y3 ϭ Ϫ4y3 Ϫ (Ϫ64)y3 ϭ Ϫ4y3 ϩ 64y3 ϭ 60y3 a6 (D) (2aϪ3b2)Ϫ2 ϭ 2Ϫ2a6bϪ4 ϭ 4b4MATCHED PROBLEM 2 Simplify using exponent properties, and express answers using positive exponents only. 9yϪ7 (A) (5xϪ3)(3x4) (B) (C) 2x4 Ϫ (Ϫ2x)4 (D) (3x4yϪ3)Ϫ2 6yϪ4 Z Scientiﬁc Notation Scientiﬁc work often involves the use of very large numbers or very small numbers. For example, the average cell contains about 200,000,000,000,000 molecules, and the diameter of an electron is about 0.000 000 000 0004 centimeter. It is generally troublesome to write and work with numbers of this type in standard decimal form. The two numbers written here cannot even be entered into most calculators as they are written. However, each can be expressed as the product of a number between 1 and 10 and an integer power of 10: 200,000,000,000,000 ϭ 2 ϫ 1014 0.000 000 000 0004 ϭ 4 ϫ 10Ϫ13 In fact, any positive number written in decimal form can be expressed in scientiﬁc nota- tion, that is, in the form a ϫ 10n 1 Յ a 6 10, n an integer, a in decimal formEXAMPLE 3 Scientiﬁc Notation (A) Write each number in scientiﬁc notation: 6,430; 5,350,000; 0.08; 0.000 32 (B) Write in standard decimal form: 2.7 ϫ 102; 9.15 ϫ 104; 5 ϫ 10Ϫ3; 8.4 ϫ 10Ϫ5 SOLUTIONS (A) 6,430 ϭ 6.43 ϫ 103; 5,350,000 ϭ 5.35 ϫ 106; 0.08 ϭ 8 ϫ 10Ϫ2; 0.000 32 ϭ 3.2 ϫ 10Ϫ4 (B) 270; 91,500; 0.005; 0.000 084MATCHED PROBLEM 3 (A) Write each number in scientiﬁc notation: 23,000; 345,000,000; 0.0031; 0.000 000 683 (B) Write in standard decimal form: 4 ϫ 103; 5.3 ϫ 105; 2.53 ϫ 10Ϫ2; 7.42 ϫ 10Ϫ6 Most calculators express very large and very small numbers in scientiﬁc notation. Con- sult the manual for your calculator to see how numbers in scientiﬁc notation are entered in your calculator. Some common methods for displaying scientiﬁc notation on a calculator are shown here. Typical Scientiﬁc Typical Graphing Number Represented Calculator Display Calculator Display 5.427 493 ϫ 10Ϫ17 5.427493 – 17 5.427493E – 17 2.359 779 ϫ 1012 2.359779 12 2.359779E12
  9. 9. 14 CHAPTER R BASIC ALGEBRAIC OPERATIONS EXAMPLE 4 Using Scientiﬁc Notation on a Calculator 325,100,000,000 Calculate by writing each number in scientiﬁc notation and then 0.000 000 000 000 0871 using your calculator. (Refer to the user’s manual accompanying your calculator for the pro- cedure.) Express the answer to three signiﬁcant digits* in scientiﬁc notation. SOLUTION 325,100,000,000 3.251 ϫ 1011 ϭ 0.000 000 000 000 0871 8.71 ϫ 10Ϫ14 ϭ 3.732491389E24 Calculator display ϭ 3.73 ϫ 10 24 To three signiﬁcant digits Figure 1 shows two solutions to this problem on a graphing calculator. In the ﬁrst solution we entered the numbers in scientiﬁc notation, and in the second we used standard decimal nota- tion. Although the multiple-line screen display on a graphing calculator enables us to enter very long standard decimals, scientiﬁc notation is usually more efﬁcient and less prone to errors in data entry. Furthermore, as Figure 1 shows, the calculator uses scientiﬁc notation toZ Figure 1 display the answer, regardless of the manner in which the numbers are entered. MATCHED PROBLEM 4 Repeat Example 4 for: 0.000 000 006 932 62,600,000,000 Z Roots of Real Numbers The solutions of the equation x2 ϭ 64 are called square roots of 64 and the solutions of x3 ϭ 64 are the cube roots of 64. So there are two real square roots of 64 (Ϫ8 and 8) and one real cube root of 64 (4 is a cube root, but Ϫ4 is not). Note that Ϫ64 has no real square root (x2 ϭ Ϫ64 has no real solution because the square of a real number can’t be negative), but Ϫ4 is a cube root of Ϫ64 because (Ϫ4)3 ϭ Ϫ64. In general: Z DEFINITION 2 Deﬁnition of an nth Root For a natural number n and a and b real numbers: a is an nth root of b if an ϭ b 3 is a fourth root of 81, since 34 ‫.18 ؍‬ The number of real nth roots of a real number b is either 0, 1, or 2, depending on whether b is positive or negative, and whether n is even or odd. Theorem 2 gives the details, which are summarized in Table 1. *For those not familiar with the meaning of signiﬁcant digits, see Appendix A for a brief discussion of this concept.
  10. 10. SECTION R–2 Exponents and Radicals 15Table 1 Number of Real Z THEOREM 2 Number of Real nth Roots of a Real Number bnth Roots of b n even n odd Let n be a natural number and let b be a real number: 1. b 7 0: If n is even, then b has two real nth roots, each the negative of theb 7 0 2 1 other; if n is odd, then b has one real nth root.bϭ0 1 1 2. b ϭ 0: 0 is the only nth root of b ϭ 0.b 6 0 0 1 3. b 6 0: If n is even, then b has no real nth root; if n is odd, then b has one real nth root. Z Rational Exponents and Radicals To denote nth roots, we can use rational exponents or we can use radicals. For example, the square root of a number b can be denoted by b1ր2 or 1b. To avoid ambiguity, both expres- sions denote the positive square root when there are two real square roots. Furthermore, both expressions are undeﬁned when there is no real square root. In general: Z DEFINITION 3 Principal nth Root For n a natural number and b a real number, the principal nth root of b, denoted by b1րn or 1b, is: n 1. The real nth root of b if there is only one. 2. The positive nth root of b if there are two real nth roots. 3. Undeﬁned if b has no real nth root. n In the notation 1b, the symbol 1 is called a radical, n is called the index, and b is the 2 radicand. If n ϭ 2, we write 1b in place of 1b. EXAMPLE 5 Principal nth Roots Evaluate each expression: (A) 91ր2 (B) 1121 3 (C) 1Ϫ125 (D) (Ϫ16)1ր4 (E) 271ր3 5 (F) 132 SOLUTIONS (A) 91ր2 ϭ 3 (B) 1121 ϭ 11 (D) (Ϫ16)1ր4 is undeﬁned (not a real number). 3 (C) 1Ϫ125 ϭ Ϫ5 (E) 271ր3 ϭ 3 5 (F) 132 ϭ 2 MATCHED PROBLEM 5 Evaluate each expression: (A) 81ր3 (D) (Ϫ1)1ր5 (F) 01ր8 4 3 (B) 1Ϫ4 (C) 110,000 (E) 1Ϫ27
  11. 11. 16 CHAPTER R BASIC ALGEBRAIC OPERATIONS How should a symbol such as 72ր3 be deﬁned? If the properties of exponents are to hold for rational exponents, then 72ր3 ϭ (71ր3)2; that is, 72ր3 must represent the square of the cube root of 7. This leads to the following general deﬁnition: Z DEFINITION 4 b m͞n and b؊m͞n, Rational Number Exponent For m and n natural numbers and b any real number (except b cannot be negative when n is even): 1 bmրn ϭ (b1րn)m and bϪmրn ϭ bm ր n 1 1 43ր2 ‫14( ؍‬ր2)3 ‫8 ؍ 32 ؍‬ 4؊3ր2 ‫؍‬ ‫؍‬ (؊4)3ր2 is not real 43ր2 8 (؊32)3ր5 ‫1)23؊([ ؍‬ր5 ] 3 ‫8؊ ؍ 3)2؊( ؍‬ We have now discussed bmրn for all rational numbers mրn and real numbers b. It can be shown, though we will not do so, that all ﬁve properties of exponents listed in Theorem 1 continue to hold for rational exponents as long as we avoid even roots of negative numbers. With the latter restriction in effect, the following useful relationship is an immediate con- sequence of the exponent properties: Z THEOREM 3 Rational Exponent/Radical Property For m and n natural numbers and b any real number (except b cannot be negative when n is even): (b1րn)m ϭ (bm)1րn n n and ( 1b)m ϭ 2bmZZZ EXPLORE-DISCUSS 1 Find the contradiction in the following chain of equations: Ϫ1 ϭ (Ϫ1)2ր2 ϭ 3(Ϫ1)2 4 1ր2 ϭ 11ր2 ϭ 1 (1) Where did we try to use Theorem 3? Why was this not correct? EXAMPLE 6 Using Rational Exponents and Radicals Simplify and express answers using positive exponents only. All letters represent positive real numbers. 4x1ր3 1ր2 (A) 82ր3 4 3 (B) 2312 (C) (3 1 x)(2 1x) (D) a b x1ր2 SOLUTIONS (A) 82ր3 ϭ (81ր3)2 ϭ 22 ϭ 4 or 82ր3 ϭ (82)1ր3 ϭ 641ր3 ϭ 4 4 12 (B) 23 ϭ (3 ) ϭ 3 ϭ 27 12 1/4 3 (C) (3 1x)(2 1x) ϭ (3x1ր3)(2x1ր2) ϭ 6x1ր3ϩ1ր2 ϭ 6x5ր6 3 4x1ր3 1ր2 41ր2x1ր6 2 2 (D) a 1ր2 b ϭ 1ր4 ϭ 1ր4Ϫ1ր6 ϭ 1ր12 x x x x
  12. 12. SECTION R–2 Exponents and Radicals 17MATCHED PROBLEM 6 Simplify and express answers using positive exponents only. All letters represent positive real numbers. 8x1ր2 1ր3 (A) (Ϫ8)5ր3 5 (B) 2324 4 3 (C) (5 2y3)(2 1 y) (D) a b x2ր3 Z Simplifying Radicals The exponent properties considered earlier lead to the following properties of radicals. Z THEOREM 4 Properties of Radicals For n a natural number greater than 1, and x and y positive real numbers: n 1. 2xn ϭ x 2 x3 ϭ x 3 n n n 2. 2xy ϭ 2x2y 2 xy ϭ 2 x2 y 5 5 5 n x n 2x x 1x 4 3. ϭ n 4 ϭ 4 By 2y Ay 1y An algebraic expression that contains radicals is said to be in simpliﬁed form if all four of the conditions listed in the following deﬁnition are satisﬁed. Z DEFINITION 5 Simpliﬁed (Radical) Form 1. No radicand (the expression within the radical sign) contains a factor to a power greater than or equal to the index of the radical. For example, 2x5 violates this condition. 2. No power of the radicand and the index of the radical have a common factor other than 1. For example, 2 x4 violates this condition. 6 3. No radical appears in a denominator. For example, y/ 1x violates this condition. 4. No fraction appears within a radical. For example, 23 violates this condition. 5EXAMPLE 7 Finding Simpliﬁed Form Write in simpliﬁed radical form. 6 6 8x4 3 (A) 212x5y2 (B) 216x4y2 (C) (D) 12x B y
  13. 13. 18 CHAPTER R BASIC ALGEBRAIC OPERATIONS SOLUTIONS (A) Condition 1 is violated. First we convert to rational exponent form. 212x5y2 ϭ (12x5y2)1ր2 Use (ab)m ‫ ؍‬ambm and (an)m ‫ ؍‬amn. ϭ 121ր2x5ր2y 12 ‫ ,3 ؒ 4 ؍‬x5ր2 ϭ x2x1ր2 ϭ (4 ؒ 3)1ր2x2x1ր2y Write in radical form. ϭ 2 13 x2 1x y Use commutative property and radical property 2. ϭ 2x2y 13x (B) Condition 2 is violated. First we convert to rational exponent form. 216x4y2 ϭ (16x4y2)1ր6 6 Use (ab)m ‫ ؍‬ambm and (an)m ‫ ؍‬amn. 1ր6 2ր3 1ր3 ϭ 16 x y 16 ‫42 ؍‬ ϭ 22ր3x2ր3y1ր3 Write in radical form. 3 ϭ 24x y 2 (C) Condition 3 is violated. We multiply numerator and denominator by 12x; the effect is to multiply the expression by 1, so its value is unchanged, but the denominator is left free of radicals. 6 6 12x 6 12x 3 12x ϭ ؒ ϭ ϭ 12x 12x 12x 2x x (D) Condition 4 is violated. First we convert to rational exponent form. 8x43 81ր3x4ր3 y2ր3 ϭ Multiply by ‫.1 ؍‬ B y y1ր3 y2ր3 2x4ր3y2ր3 ϭ x 4ր3 ‫ ؍‬xx 1ր3 y 2xx1ր3y2ր3 ϭ Write in radical form. y 3 2x 2xy2 ϭ y MATCHED PROBLEM 7 Write in simpliﬁed radical form. 9 30 5x3 (A) 218x4y3 (B) 28x6y3 (C) (D) 4 1 16x B y Eliminating a radical from a denominator [as in Example 7(C)] is called rationalizing the denominator. To rationalize the denominator, we multiply the numerator and denomi- nator by a suitable factor that will leave the denominator free of radicals. This factor is called a rationalizing factor. If the denominator is of the form 1a ϩ 1b, then 1a Ϫ 1b is a rationalizing factor because ( 1a ϩ 1b)(1a Ϫ 1b) ϭ a Ϫ b Similarly, if the denominator is of the form 1a Ϫ 1b, then 1a ϩ 1b is a rationalizing factor. EXAMPLE 8 Rationalizing Denominators Rationalize the denominator and write the answer in simpliﬁed radical form. 8 1x ϩ 1y (A) (B) 16 ϩ 15 1x Ϫ 1y
  14. 14. SECTION R–2 Exponents and Radicals 19 SOLUTIONS (A) Multiply numerator and denominator by the rationalizing factor 16 Ϫ 15. 8 8 16 Ϫ 15 ϭ ؒ ( 1a ؉ 1b) ( 1a ؊ 1b) ‫ ؍‬a ؊ b 16 ϩ 15 16 ϩ 15 16 Ϫ 15 8( 16 Ϫ 15) ϭ Simplify. 6Ϫ5 ϭ 8( 16 Ϫ 15) (B) Multiply numerator and denominator by the rationalizing factor 1x ϩ 1y. 1x ϩ 1y 1x ϩ 1y 1x ϩ 1y ϭ ؒ Expand numerator and denominator. 1x Ϫ 1y 1x Ϫ 1y 1x ϩ 1y x ϩ 1x1y ϩ 1y1x ϩ y ϭ Combine like terms. xϪy x ϩ 2 1xy ϩ y ϭ xϪy MATCHED PROBLEM 8 Rationalize the denominator and write the answer in simpliﬁed radical form. 6 2 1x Ϫ 3 1y (A) (B) 1 Ϫ 13 1x ϩ 1y ANSWERS TO MATCHED PROBLEMS 1. (A) 1 (B) 0.000 01 (C) x4 (D) v3րu7 2. (A) 15x (B) 3ր(2y ) 3 (C) Ϫ14x 4 (D) y6 ր(9x8) 3. (A) 2.3 ϫ 10 ; 3.45 ϫ 10 ; 3.1 ϫ 10 ; 6.83 ϫ 10Ϫ7 4 8 Ϫ3 (B) 4,000; 530,000; 0.0253; 0.000 007 42 4. 1.11 ϫ 10Ϫ19 5. (A) 2 (B) Not real (C) 10 (D) Ϫ1 (E) Ϫ3 (F) 0 6. (A) Ϫ32 (B) 16 (C) 10y13ր12 (D) 2 րx1ր18 152x34 x 15xy 7. (A) 3x2y12y 3 (B) 22x2y (C) (D) x y 2x Ϫ 5 1xy ϩ 3y 8. (A) Ϫ3 Ϫ 313 (B) xϪy R-2 ExercisesAll variables represent positive real numbers and are restricted to 7. (Ϫ5)4 8. (Ϫ4)5 9. (Ϫ3)Ϫ1prevent division by 0. 10. (Ϫ7)Ϫ2 11. Ϫ7Ϫ2 12. Ϫ100In Problems 1–14, evaluate each expression. If the answer is notan integer, write it in fraction form. 1 0 1 Ϫ1 13. a b 14. a b 1 8 3 10 1. 37 2. 56 3. a b 2 3 3 Ϫ3 Ϫ6 4. a b 5. 6 6. 2 5
  15. 15. 20 CHAPTER R BASIC ALGEBRAIC OPERATIONSIn Problems 15–20, write the numbers in scientiﬁc notation. 1 1 3 75. 76. 77. 17 315. 58,620,000 16. 4,390 215 3 25417. 0.027 18. 0.11 12y2 4 12 78. 79. 80. 16y 16 Ϫ 2 16 ϩ 219. 0.000 000 064 20. 0.000 0325 5 3 12m15 81. x 236x7y11 82. 2a28a8b13 83.In Problems 21–26, write each number in standard decimal form. 120m21. 4 ϫ 10Ϫ3 22. 5 ϫ 10Ϫ6 31y 215 ϩ 3 12 312 Ϫ 2 13 84. 85. 86. 21y Ϫ 3 515 ϩ 2 12 313 Ϫ 2 1223. 2.99 ϫ 105 24. 7.75 ϫ 1011 225. 3.1 ϫ 10Ϫ7 26. 8.167 ϫ 10Ϫ4 87. What is the result of entering 23 on a calculator? 2 88. Refer to Problem 87. What is the difference between 2(3 ) and 2In Problems 27–32, change to radical form. Do not simplify. (23)2? Which agrees with the value of 23 obtained with a calcu- lator?27. 321ր5 28. 6253ր4 29. 4xϪ1ր230. 32yϪ2ր5 31. x1ր3 Ϫ y1ր3 32. (x Ϫ y)1ր3 APPLICATIONSIn Problems 33–38, change to rational exponent form. Do not 89. ECONOMICS If in the United States in 2007 the national debtsimplify. was about \$8,868,000,000,000 and the population was about 3 533. 1361 34. 2172 35. 4x2y3 301,000,000, estimate to three signiﬁcant digits each individual’s 4 3 3 3 share of the national debt. Write your answer in scientiﬁc notation36. 27x3y2 37. 2x2 ϩ y2 38. 2x2 ϩ 2y2 and in standard decimal form. 90. ECONOMICS If in the United States in 2007 the gross domestic product (GDP) was about \$14,074,000,000,000 and the populationIn Problems 39–50, evaluate each expression that represents a was about 301,000,000, estimate to three signiﬁcant digits the GDPreal number. per person. Write your answer in scientiﬁc notation and in standard39. 1001ր2 40. 1691ր2 decimal form.41. 1121 42. 1361 91. ECONOMICS The number of units N of a ﬁnished product pro- 1ր3 2ր3 duced from the use of x units of labor and y units of capital for a43. 125 44. 27 particular Third World country is approximated by45. 1Ϫ27 3 46. 164 3 N ϭ 10x3ր4y1ր447. 1Ϫ16 4 48. 1Ϫ1 6 Estimate how many units of a ﬁnished product will be produced us- Ϫ3ր249. 9 50. 64Ϫ4ր3 ing 256 units of labor and 81 units of capital. 92. ECONOMICS The number of units N of a ﬁnished product pro-In Problems 51–64, simplify and express answers using positive duced by a particular automobile company where x units of laborexponents only. and y units of capital are used is approximated by51. x5xϪ2 52. y6yϪ8 53. (2y)(3y2)(5y4) N ϭ 50x1ր2y1ր254. (6x3)(4x7)(xϪ5) 55. (a2b3)5 56. (2c4dϪ2)Ϫ3 Estimate how many units will be produced using 256 units of labor 1ր3 5ր3 Ϫ1ր5 6ր5 Ϫ3 1ր6 and 144 units of capital.57. u u 58. v v 59. (x ) 93. BRAKING DISTANCE R. A. Moyer of Iowa State College found, mϪ2n3 2 6mnϪ2 Ϫ360. (49a4bϪ2)1ր2 61. a b 62. a b in comprehensive tests carried out on 41 wet pavements, that the m4nϪ1 3mϪ1n2 braking distance d (in feet) for a particular automobile traveling at w4 Ϫ1ր2 8aϪ4b3 1ր3 v miles per hour was given approximately by63. a b 64. a b 9xϪ2 27a2bϪ3 d ϭ 0.0212v7ր3 Approximate the braking distance to the nearest foot for the carIn Problems 65–86, write in simpliﬁed radical form. traveling on wet pavement at 70 miles per hour.65. Ϫ1128 66. Ϫ1125 94. BRAKING DISTANCE Approximately how many feet would it take the car in Problem 93 to stop on wet pavement if it were trav-67. 127 Ϫ 513 68. 218 ϩ 118 eling at 50 miles per hour? (Compute answer to the nearest foot.)69. 25 Ϫ 225 ϩ 2625 3 3 3 70. 220 ϩ 240 Ϫ 25 3 3 3 371. 225 210 72. 16114 473. 216m4y8 74. 216m4n8
  16. 16. SECTION R–3 Polynomials: Basic Operations and Factoring 2195. PHYSICS—RELATIVISTIC MASS The mass M of an object and back again is called the period T and is given bymoving at a velocity v is given by L T ϭ 2␲ M0 Ag Mϭ v2 1Ϫ where g is the gravitational constant. Show that T can be written in A c2 the formwhere M0 ϭ mass at rest and c ϭ velocity of light. The mass of an 2␲1gLobject increases with velocity and tends to inﬁnity as the velocity Tϭ gapproaches the speed of light. Show that M can be written in theform M0c 2c2 Ϫ v2 Mϭ c2 Ϫ v296. PHYSICS—PENDULUM A simple pendulum is formed by hang-ing a bob of mass M on a string of length L from a ﬁxed support(see the ﬁgure). The time it takes the bob to swing from right to left R-3 Polynomials: Basic Operations and Factoring Z Polynomials Z Addition and Subtraction Z Multiplication Z Factoring In this section, we review the basic operations on polynomials. Polynomials are expres- sions such as x4 Ϫ 5x2 ϩ 1 or 3xy Ϫ 2x ϩ 5y ϩ 6 that are built from constants and vari- ables using only addition, subtraction, and multiplication (the power x4 is the product x ؒ x ؒ x ؒ x). Polynomials are used throughout mathematics to describe and approximate mathematical relationships. Z Polynomials Algebraic expressions are formed by using constants and variables and the algebraic oper- ations of addition, subtraction, multiplication, division, raising to powers, and taking roots. Some examples are 3 2x3 ϩ 5 5x4 ϩ 2x2 Ϫ 7 xϪ5 1 1ϩ x ϩ 2x Ϫ 5 2 1 1ϩ x An algebraic expression involving only the operations of addition, subtraction, multi- plication, and raising to natural number powers is called a polynomial. (Note that raising to a natural number power is repeated multiplication.) Some examples are 2x Ϫ 3 4x2 Ϫ 3x ϩ 7 x Ϫ 2y x3 Ϫ 3x2y ϩ xy2 ϩ 2y7
  17. 17. 22 CHAPTER R BASIC ALGEBRAIC OPERATIONS In a polynomial, a variable cannot appear in a denominator, as an exponent, or within a rad- ical. Accordingly, a polynomial in one variable x is constructed by adding or subtracting constants and terms of the form axn, where a is a real number and n is a natural number. A polynomial in two variables x and y is constructed by adding and subtracting constants and terms of the form axmyn, where a is a real number and m and n are natural numbers. Polynomials in three or more variables are deﬁned in a similar manner. Polynomials can be classiﬁed according to their degree. If a term in a polynomial has only one variable as a factor, then the degree of that term is the power of the variable. If two or more variables are present in a term as factors, then the degree of the term is the sum of the powers of the variables. The degree of a polynomial is the degree of the nonzero term with the highest degree in the polynomial. Any nonzero constant is deﬁned to be a polynomial of degree 0. The number 0 is also a polynomial but is not assigned a degree. EXAMPLE 1 Polynomials and Nonpolynomials (A) Which of the following are polynomials? 1 2x ϩ 5 Ϫ x2 Ϫ 3x ϩ 2 2x3 Ϫ 4x ϩ 1 x4 ϩ 12 x (B) Given the polynomial 2x3 Ϫ x6 ϩ 7, what is the degree of the ﬁrst term? The third term? The whole polynomial? (C) Given the polynomial x3y2 ϩ 2x2y ϩ 1, what is the degree of the ﬁrst term? The sec- ond term? The whole polynomial? SOLUTIONS (A) x2 Ϫ 3x ϩ 2 and x4 ϩ 12 are polynomials. (The others are not polynomials since a variable appears in a denominator or within a radical.) (B) The ﬁrst term has degree 3, the third term has degree 0, and the whole polynomial has degree 6. (C) The ﬁrst term has degree 5, the second term has degree 3, and the whole polynomial has degree 5. MATCHED PROBLEM 1 (A) Which of the following are polynomials? xϪ1 3x2 Ϫ 2x ϩ 1 1x Ϫ 3 x2 Ϫ 2xy ϩ y2 x2 ϩ 2 (B) Given the polynomial 3x5 Ϫ 6x3 ϩ 5, what is the degree of the ﬁrst term? The second term? The whole polynomial? (C) Given the polynomial 6x4y2 Ϫ 3xy3, what is the degree of the ﬁrst term? The second term? The whole polynomial? In addition to classifying polynomials by degree, we also call a single-term polynomial a monomial, a two-term polynomial a binomial, and a three-term polynomial a trinomial. 5 2 3 2x y Monomial x ϩ 4.7 3 Binomial x4 Ϫ 12x2 ϩ 9 Trinomial A constant in a term of a polynomial, including the sign that precedes it, is called the numerical coefﬁcient, or simply, the coefﬁcient, of the term. If a constant doesn’t appear, or
  18. 18. SECTION R–3 Polynomials: Basic Operations and Factoring 23 only a ϩ sign appears, the coefﬁcient is understood to be 1. If only a Ϫ sign appears, the coef- ﬁcient is understood to be Ϫ1. So given the polynomial 2x4 Ϫ 4x3 ϩ x2 Ϫ x ϩ 5 2x4 ؉ (؊4)x3 ؉ 1x2 ؉ (؊1)x ؉ 5 the coefﬁcient of the ﬁrst term is 2, the coefﬁcient of the second term is Ϫ4, the coefﬁ- cient of the third term is 1, the coefﬁcient of the fourth term is Ϫ1, and the coefﬁcient of the last term is 5. Two terms in a polynomial are called like terms if they have exactly the same variable factors to the same powers. The numerical coefﬁcients may or may not be the same. Since constant terms involve no variables, all constant terms are like terms. If a polynomial con- tains two or more like terms, these terms can be combined into a single term by making use of distributive properties. Consider the following example: 5x3y Ϫ 2xy Ϫ x3y Ϫ 2x3y ϭ 5x3y Ϫ x3y Ϫ 2x3y Ϫ 2xy Group like terms. ϭ (5x3y Ϫ x3y Ϫ 2 x3y) Ϫ 2xy Use the distributive property ϭ (5 Ϫ 1 Ϫ 2) x y Ϫ 2xy 3 Simplify. ϭ 2x3y Ϫ 2xy It should be clear that free use has been made of the real number properties discussed earlier. The steps done in the dashed box are usually done mentally, and the process is quickly done as follows: Like terms in a polynomial are combined by adding their numerical coefﬁcients. Z Addition and Subtraction Addition and subtraction of polynomials can be thought of in terms of removing paren- theses and combining like terms. Horizontal and vertical arrangements are illustrated in the next two examples. You should be able to work either way, letting the situation dic- tate the choice.EXAMPLE 2 Adding Polynomials Add: x4 Ϫ 3x3 ϩ x2, Ϫx3 Ϫ 2x2 ϩ 3x, and 3x2 Ϫ 4x Ϫ 5 SOLUTION Add horizontally: (x4 Ϫ 3x3 ϩ x2) ϩ (Ϫx3 Ϫ 2x2 ϩ 3x) ϩ (3x2 Ϫ 4x Ϫ 5) Remove parentheses. ϭ x Ϫ 3x ϩ x Ϫ x Ϫ 2x ϩ 3x ϩ 3x Ϫ 4x Ϫ 5 4 3 2 3 2 2 Combine like terms. ϭ x4 Ϫ 4x3 ϩ 2x2 Ϫ x Ϫ 5 Or vertically, by lining up like terms and adding their coefﬁcients: x4 Ϫ 3x3 ϩ x2 Ϫ x3 Ϫ 2x2 ϩ 3x 3x2 Ϫ 4x Ϫ 5 x Ϫ 4x ϩ 2x2 Ϫ x Ϫ 5 4 3MATCHED PROBLEM 2 Add horizontally and vertically: 3x4 Ϫ 2x3 Ϫ 4x2, x3 Ϫ 2x2 Ϫ 5x, and x2 ϩ 7x Ϫ 2
  19. 19. 24 CHAPTER R BASIC ALGEBRAIC OPERATIONS EXAMPLE 3 Subtracting Polynomials Subtract: 4x2 Ϫ 3x ϩ 5 from x2 Ϫ 8 SOLUTION (x2 Ϫ 8) Ϫ (4x2 Ϫ 3x ϩ 5) or x2 Ϫ 8 ϭ x2 Ϫ 8 Ϫ 4x2 ϩ 3x Ϫ 5 Ϫ4x ϩ 3x Ϫ 5 2 d Change signs and add. ϭ Ϫ3x2 ϩ 3x Ϫ 13 Ϫ3x2 ϩ 3x Ϫ 13 MATCHED PROBLEM 3 Subtract: 2x2 Ϫ 5x ϩ 4 from 5x2 Ϫ 6 ZZZ CAUTION Z Z Z When you use a horizontal arrangement to subtract a polynomial with more than one term, you must enclose the polynomial in parentheses. For example, to subtract 2x ϩ 5 from 4x Ϫ 11, you must write 4x Ϫ 11 Ϫ (2x ϩ 5) and not 4x Ϫ 11 Ϫ 2x ϩ 5 Z Multiplication Multiplication of algebraic expressions involves extensive use of distributive properties for real numbers, as well as other real number properties. EXAMPLE 4 Multiplying Polynomials Multiply: (2x Ϫ 3)(3x2 Ϫ 2x ϩ 3) SOLUTION (2x Ϫ 3)(3x2 Ϫ 2x ϩ 3) ϭ 2x(3x2 Ϫ 2x ϩ 3) Ϫ 3(3x2 Ϫ 2x ϩ 3) Distribute, multiply out parentheses. ϭ 6x3 Ϫ 4x2 ϩ 6x Ϫ 9x2 ϩ 6x Ϫ 9 Combine like terms. ϭ 6x Ϫ 13x ϩ 12x Ϫ 9 3 2 Or, using a vertical arrangement, 3x2 Ϫ 2x ϩ 3 2x Ϫ 3 6x3 Ϫ 4x2 ϩ 6x Ϫ 9x2 ϩ 6x Ϫ 9 6x3 Ϫ 13x2 ϩ 12x Ϫ 9 MATCHED PROBLEM 4 Multiply: (2x Ϫ 3)(2x2 ϩ 3x Ϫ 2) To multiply two polynomials, multiply each term of one by each term of the other, and combine like terms.
  20. 20. SECTION R–3 Polynomials: Basic Operations and Factoring 25Z FactoringA factor of a number is one of two or more numbers whose product is the given number.Similarly, a factor of an algebraic expression is one of two or more algebraic expressionswhose product is the given algebraic expression. For example, 30 ϭ 2 ؒ 3 ؒ 5 2, 3, and 5 are each factors of 30. x Ϫ 4 ϭ (x Ϫ 2)(x ϩ 2) 2 (x ؊ 2) and (x ؉ 2) are each factors of x2 ؊ 4.The process of writing a number or algebraic expression as the product of other numbersor algebraic expressions is called factoring. We start our discussion of factoring with thepositive integers. An integer such as 30 can be represented in a factored form in many ways. The products 6ؒ5 (1)(10)(6) 2 15 ؒ 2 2ؒ3ؒ5all yield 30. A particularly useful way of factoring positive integers greater than 1 is interms of prime numbers. An integer greater than 1 is prime if its only positive integer factors are itself and 1.So 2, 3, 5, and 7 are prime, but 4, 6, 8, and 9 are not prime. An integer greater than 1 thatis not prime is called a composite number. The integer 1 is neither prime nor composite. A composite number is said to be factored completely if it is represented as a prod-uct of prime factors. The only factoring of 30 that meets this condition, except for the orderof the factors, is 30 ϭ 2 ؒ 3 ؒ 5. This illustrates an important property of integers. Z THEOREM 1 The Fundamental Theorem of Arithmetic Each integer greater than 1 is either prime or can be expressed uniquely, except for the order of factors, as a product of prime factors. We can also write polynomials in completely factored form. A polynomial such as2x2 Ϫ x Ϫ 6 can be written in factored form in many ways. The products (2x ϩ 3)(x Ϫ 2) 2(x2 Ϫ 1x Ϫ 3) 2 2(x ϩ 3)(x Ϫ 2) 2all yield 2x2 Ϫ x Ϫ 6. A particularly useful way of factoring polynomials is in terms ofprime polynomials. Z DEFINITION 1 Prime Polynomials A polynomial of degree greater than 0 is said to be prime relative to a given set of numbers if: (1) all of its coefﬁcients are from that set of numbers; and (2) it cannot be written as a product of two polynomials (excluding constant polynomials that are factors of 1) having coefﬁcients from that set of numbers. Relative to the set of integers: x2 ؊ 2 is prime x2 ؊ 9 is not prime, since x2 ؊ 9 ‫( ؍‬x ؊ 3)(x ؉ 3) [Note: The set of numbers most frequently used in factoring polynomials is the set of integers.]
  21. 21. 26 CHAPTER R BASIC ALGEBRAIC OPERATIONS A nonprime polynomial is said to be factored completely relative to a given set of numbers if it is written as a product of prime polynomials relative to that set of numbers. In Examples 5 and 6 we review some of the standard factoring techniques for polyno- mials with integer coefﬁcients. EXAMPLE 5 Factoring Out Common Factors Factor out, relative to the integers, all factors common to all terms: (A) 2x3y Ϫ 8x2y2 Ϫ 6xy3 (B) 2x(3x Ϫ 2) Ϫ 7(3x Ϫ 2) SOLUTIONS (A) 2x3y Ϫ 8x2y2 Ϫ 6xy3 ϭ (2xy)x2 Ϫ (2xy)4xy Ϫ (2xy)3y2 Factor out 2xy. ϭ 2xy(x2 Ϫ 4xy Ϫ 3y2) (B) 2x(3x Ϫ 2) Ϫ 7(3x Ϫ 2) ϭ 2x(3x Ϫ 2) Ϫ 7(3x Ϫ 2) Factor out 3x ؊ 2. ϭ (2x Ϫ 7)(3x Ϫ 2) MATCHED PROBLEM 5 Factor out, relative to the integers, all factors common to all terms: (A) 3x3y Ϫ 6x2y2 Ϫ 3xy3 (B) 3y(2y ϩ 5) ϩ 2(2y ϩ 5) The polynomials in Example 6 can be factored by ﬁrst grouping terms to ﬁnd a com- mon factor. EXAMPLE 6 Factoring by Grouping Factor completely, relative to the integers, by grouping: (A) 3x2 Ϫ 6x ϩ 4x Ϫ 8 (B) wy ϩ wz Ϫ 2xy Ϫ 2xz (C) 3ac ϩ bd Ϫ 3ad Ϫ bc SOLUTIONS (A) 3x2 Ϫ 6x ϩ 4x Ϫ 8 Group the ﬁrst two and last two terms. ϭ (3x2 Ϫ 6x) ϩ (4x Ϫ 8) Remove common factors from each group. ϭ 3x(x Ϫ 2) ϩ 4(x Ϫ 2) Factor out the common factor (x ؊ 2). ϭ (3x ϩ 4)(x Ϫ 2) (B) wy ϩ wz Ϫ 2xy Ϫ 2xz Group the ﬁrst two and last two terms—be careful of signs. ϭ (wy ϩ wz) Ϫ (2xy ϩ 2xz) Remove common factors from each group. ϭ w( y ϩ z) Ϫ 2x(y ϩ z) Factor out the common factor ( y ؉ z). ϭ (w Ϫ 2x)(y ϩ z) (C) 3ac ϩ bd Ϫ 3ad Ϫ bc In parts (A) and (B) the polynomials are arranged in such a way that grouping the ﬁrst two terms and the last two terms leads to common factors. In this problem nei- ther the ﬁrst two terms nor the last two terms have a common factor. Sometimes rearranging terms will lead to a factoring by grouping. In this case, we interchange
  22. 22. SECTION R–3 Polynomials: Basic Operations and Factoring 27 the second and fourth terms to obtain a problem comparable to part (B), which can be factored as follows: 3ac Ϫ bc Ϫ 3ad ϩ bd ϭ (3ac Ϫ bc) Ϫ (3ad Ϫ bd) Factor out c, d. ϭ c(3a Ϫ b) Ϫ d(3a Ϫ b) Factor out 3a ؊ b. ϭ (c Ϫ d)(3a Ϫ b)MATCHED PROBLEM 6 Factor completely, relative to the integers, by grouping: (A) 2x2 ϩ 6x ϩ 5x ϩ 15 (B) 2pr ϩ ps Ϫ 6qr Ϫ 3qs (C) 6wy Ϫ xz Ϫ 2xy ϩ 3wz Example 7 illustrates an approach to factoring a second-degree polynomial of the form 2x2 Ϫ 5x Ϫ 3 or 2x2 ϩ 3xy Ϫ 2y2 into the product of two ﬁrst-degree polynomials with integer coefﬁcients.EXAMPLE 7 Factoring Second-Degree Polynomials Factor each polynomial, if possible, using integer coefﬁcients: (A) 2x2 ϩ 3xy Ϫ 2y2 (B) x2 Ϫ 3x ϩ 4 (C) 6x2 ϩ 5xy Ϫ 4y2 SOLUTIONS (A) 2x2 ϩ 3xy Ϫ 2y2 ϭ (2x ϩ y)(x Ϫ y) Put in what we know. Signs must be opposite. (We can reverse this choice c c if we get ؊3xy instead of ؉3xy for ? ? the middle term.) Now, what are the factors of 2 (the coefﬁcient of y2)? 2 1ؒ2 (2x ؉ y)(x ؊ 2y) ‫2 ؍‬x2 ؊ 3xy ؊ 2y2 2ؒ1 (2x ؉ 2y)(x ؊ y) ‫2 ؍‬x2 ؊ 2y2 The ﬁrst choice gives us Ϫ3xy for the middle term—close, but not there—so we reverse our choice of signs to obtain 2x2 ϩ 3xy Ϫ 2y2 ϭ (2x Ϫ y)(x ϩ 2y) (B) x2 Ϫ 3x ϩ 4 ϭ (x Ϫ )(x Ϫ ) Signs must be the same because the third term is positive and must be negative because the middle term is negative. 4 2ؒ2 (x ؊ 2)(x ؊ 2) ‫ ؍‬x2 ؊ 4x ؉ 4 1ؒ4 (x ؊ 1)(x ؊ 4) ‫ ؍‬x2 ؊ 5x ؉ 4 4ؒ1 (x ؊ 4)(x ؊ 1) ‫ ؍‬x2 ؊ 5x ؉ 4 No choice produces the middle term; so x2 Ϫ 3x ϩ 4 is not factorable using integer coefﬁcients. (C) 6x2 ϩ 5xy Ϫ 4y2 ϭ ( x ϩ y)( x Ϫ y) c c c c ? ? ? ?
  23. 23. 28 CHAPTER R BASIC ALGEBRAIC OPERATIONS The signs must be opposite in the factors, because the third term is negative. We can reverse our choice of signs later if necessary. We now write all factors of 6 and of 4: 6 4 2ؒ3 2ؒ2 3ؒ2 1ؒ4 1ؒ6 4ؒ1 6ؒ1 and try each choice on the left with each on the right—a total of 12 combinations that give us the ﬁrst and last terms in the polynomial 6x2 ϩ 5xy Ϫ 4y2. The question is: Does any combination also give us the middle term, 5xy? After trial and error and, perhaps, some educated guessing among the choices, we ﬁnd that 3 ؒ 2 matched with 4 ؒ 1 gives us the correct middle term. 6x2 ϩ 5xy Ϫ 4y2 ϭ (3x ϩ 4y)(2x Ϫ y) If none of the 24 combinations (including reversing our sign choice) had produced the middle term, then we would conclude that the polynomial is not factorable using integer coefﬁcients. MATCHED PROBLEM 7 Factor each polynomial, if possible, using integer coefﬁcients: (A) x2 Ϫ 8x ϩ 12 (B) x2 ϩ 2x ϩ 5 (C) 2x2 ϩ 7xy Ϫ 4y2 (D) 4x2 Ϫ 15xy Ϫ 4y2 The special factoring formulas listed here will enable us to factor certain polynomial forms that occur frequently. Z SPECIAL FACTORING FORMULAS 1. u2 ϩ 2uv ϩ v2 ϭ (u ϩ v)2 Perfect Square 2. u Ϫ 2uv ϩ v ϭ (u Ϫ v) 2 2 2 Perfect Square 3. u Ϫ v ϭ (u Ϫ v)(u ϩ v) 2 2 Difference of Squares 4. u Ϫ v ϭ (u Ϫ v)(u ϩ uv ϩ v ) 3 3 2 2 Difference of Cubes 5. u ϩ v ϭ (u ϩ v)(u Ϫ uv ϩ v ) 3 3 2 2 Sum of Cubes The formulas in the box can be established by multiplying the factors on the right.ZZZ EXPLORE-DISCUSS 1 Explain why there is no formula for factoring a sum of squares u2 ϩ v2 into the product of two ﬁrst-degree polynomials with real coefﬁcients.
  24. 24. SECTION R–3 Polynomials: Basic Operations and Factoring 29 EXAMPLE 8 Using Special Factoring Formulas Factor completely relative to the integers: (A) x2 ϩ 6xy ϩ 9y2 (B) 9x2 Ϫ 4y2 (C) 8m3 Ϫ 1 (D) x3 ϩ y3z3 SOLUTIONS (A) x2 ϩ 6xy ϩ 9y2 ϭ x2 ϩ 2(x)(3y) ϩ (3y)2 ϭ (x ϩ 3y)2 Perfect square (B) 9x2 Ϫ 4y2 ϭ (3x)2 Ϫ (2y)2 ϭ (3x Ϫ 2y)(3x ϩ 2y) Difference of squares (C) 8m3 Ϫ 1 ϭ (2m)3 Ϫ 13 Difference of cubes ϭ (2m Ϫ 1) ΄(2m)2 ϩ (2m)(1) ϩ 12΅ Simplify. ϭ (2m Ϫ 1)(4m2 ϩ 2m ϩ 1) (D) x3 ϩ y3z3 ϭ x3 ϩ ( yz)3 Sum of cubes ϭ (x ϩ yz)(x2 Ϫ xyz ϩ y2z2) MATCHED PROBLEM 8 Factor completely relative to the integers: (A) 4m2 Ϫ 12mn ϩ 9n2 (B) x2 Ϫ 16y2 (C) z3 Ϫ 1 (D) m3 ϩ n3 ANSWERS TO MATCHED PROBLEMS 1. (A) 3x2 Ϫ 2x ϩ 1, x2 Ϫ 2xy ϩ y2 (B) 5, 3, 5 (C) 6, 4, 6 2. 3x4 Ϫ x3 Ϫ 5x2 ϩ 2x Ϫ 2 3. 3x2 ϩ 5x Ϫ 10 4. 4x3 Ϫ 13x ϩ 6 5. (A) 3xy(x2 Ϫ 2xy Ϫ y2) (B) (3y ϩ 2)(2y ϩ 5) 6. (A) (2x ϩ 5)(x ϩ 3) (B) (p Ϫ 3q)(2r ϩ s) (C) (3w Ϫ x)(2y ϩ z) 7. (A) (x Ϫ 2)(x Ϫ 6) (B) Not factorable using integers (C) (2x Ϫ y)(x ϩ 4y) (D) (4x ϩ y)(x Ϫ 4y) 8. (A) (2m Ϫ 3n)2 (B) (x Ϫ 4y)(x ϩ 4y) (C) (z Ϫ 1)(z2 ϩ z ϩ 1) (D) (m ϩ n)(m Ϫ mn ϩ n ) 2 2 R-3 ExercisesProblems 1–8 refer to the polynomials (a) x2 ϩ 1 and In Problems 9–14, is the algebraic expression a polynomial? If so,(b) x4 Ϫ 2x ϩ 1. give its degree.1. What is the degree of (a)? 9. 4 Ϫ x2 10. x3 Ϫ 5x6 ϩ 12. What is the degree of (b)? 11. x3 Ϫ 7x ϩ 81x 12. x4 ϩ 3x Ϫ 153. What is the degree of the sum of (a) and (b)? 13. x5 Ϫ 4x2 ϩ 6Ϫ2 14. 3x4 Ϫ 2xϪ1 Ϫ 104. What is the degree of the product of (a) and (b)? In Problems 15–22, perform the indicated operations and simplify.5. Multiply (a) and (b). 15. 2(x Ϫ 1) ϩ 3(2x Ϫ 3) Ϫ (4x Ϫ 5)6. Add (a) and (b). 16. 2y Ϫ 3y [4 Ϫ 2( y Ϫ 1) ] 17. (m Ϫ n)(m ϩ n)7. Subtract (b) from (a).8. Subtract (a) from (b).