Like this document? Why not share!

# Pre-calculus

## by Imran NurHakim, university student at TANTA UNIVERSITY on Feb 21, 2013

• 299 views

### Views

Total Views
299
Views on SlideShare
299
Embed Views
0

Likes
0
5
0

No embeds

## Pre-calculusDocument Transcript

• 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)
• 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
• 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.”
• 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
• 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.
• 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
• 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.
• 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
• 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.
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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.
• 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.]
• 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
• 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 ? ? ? ?
• 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.
• 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).
• 30 CHAPTER R BASIC ALGEBRAIC OPERATIONS18. (5y Ϫ 1)(3 Ϫ 2y) 19. (3x ϩ 2y)(x Ϫ 3y) 65. 2(x ϩ h)2 Ϫ 3(x ϩ h) Ϫ (2x2 Ϫ 3x)20. (4x Ϫ y)2 21. (a ϩ b)(a2 Ϫ ab ϩ b2) 66. Ϫ4(x ϩ h)2 ϩ 6(x ϩ h) Ϫ (Ϫ4x2 ϩ 6x)22. (a Ϫ b)(a2 ϩ ab ϩ b2) 67. (x ϩ h)3 Ϫ 2(x ϩ h)2 Ϫ (x3 Ϫ 2x2) 68. (x ϩ h)3 ϩ 3(x ϩ h) Ϫ (x3 ϩ 3x)In Problems 23–28, factor out, relative to the integers, all factorscommon to all terms. Problems 69–74 are calculus-related. Factor completely, relative23. 6x4 Ϫ 8x3 Ϫ 2x2 24. 3x5 ϩ 6x3 ϩ 9x to the integers.25. x2y ϩ 2xy2 ϩ x2y2 26. 8u3v Ϫ 6u2v2 ϩ 4uv3 69. 2x(x ϩ 1)4 ϩ 4x2(x ϩ 1)327. 2w( y Ϫ 2z) Ϫ x( y Ϫ 2z) 70. (x Ϫ 1)3 ϩ 3x(x Ϫ 1)228. 2x(u Ϫ 3v) ϩ 5y(u Ϫ 3v) 71. 6(3x Ϫ 5)(2x Ϫ 3)2 ϩ 4(3x Ϫ 5)2(2x Ϫ 3) 72. 2(x Ϫ 3)(4x ϩ 7)2 ϩ 8(x Ϫ 3)2(4x ϩ 7)In Problems 29–34, factor completely, relative to the integers. 73. 5x4(9 Ϫ x)4 Ϫ 4x5(9 Ϫ x)329. x ϩ 4x ϩ x ϩ 4 2 30. 2y Ϫ 6y ϩ 5y Ϫ 15 2 74. 3x4(x Ϫ 7)2 ϩ 4x3(x Ϫ 7)331. x Ϫ xy ϩ 3xy Ϫ 3y 2 2 32. 3a2 Ϫ 12ab Ϫ 2ab ϩ 8b233. 8ac ϩ 3bd Ϫ 6bc Ϫ 4ad In Problems 75–86, factor completely, relative to the integers.34. 3ux Ϫ 4vy ϩ 3vx Ϫ 4uy In polynomials involving more than three terms, try grouping the terms in various combinations as a ﬁrst step. If a polynomial is prime relative to the integers, say so.In Problems 35–42, perform the indicated operations and simplify. 75. (a Ϫ b)2 Ϫ 4(c Ϫ d )235. 2x Ϫ 35x ϩ 2 3 x Ϫ (x ϩ 5)4 ϩ 16 76. (x ϩ 2)2 ϩ 936. m Ϫ 5m Ϫ 3 m Ϫ (m Ϫ 1)4 6 77. 2am Ϫ 3an ϩ 2bm Ϫ 3bn37. (2x2 Ϫ 3x ϩ 1)(x2 ϩ x Ϫ 2) 78. 15ac Ϫ 20ad ϩ 3bc Ϫ 4bd38. (x2 Ϫ 3xy ϩ y2)(x2 ϩ 3xy ϩ y2) 79. 3x2 Ϫ 2xy Ϫ 4y239. (3u Ϫ 2v) Ϫ (2u Ϫ 3v)(2u ϩ 3v) 2 80. 5u2 ϩ 4uv Ϫ v240. (2a Ϫ b)2 Ϫ (a ϩ 2b)2 81. x3 Ϫ 3x2 Ϫ 9x ϩ 27 82. t3 Ϫ 2t 2 ϩ t Ϫ 241. (2m Ϫ n) 3 42. (3a ϩ 2b) 3 83. 4(A ϩ B)2 Ϫ 5(A ϩ B) Ϫ 5 84. x4 ϩ 6x2 ϩ 8 85. m4 Ϫ n4In Problems 43–62, factor completely, relative to the integers. If apolynomial is prime relative to the integers, say so. 86. y4 Ϫ 3y2 Ϫ 443. 2x2 ϩ x Ϫ 3 44. 3y2 Ϫ 8y Ϫ 3 87. Show by example that, in general, (a ϩ b)2 a2 ϩ b2. Dis- cuss possible conditions on a and b that would make this a45. x2 ϩ 5xy Ϫ 14y2 46. x2 ϩ 4y2 valid equation.47. 4x2 Ϫ 20x ϩ 25 48. a2b2 Ϫ c2 88. Show by example that, in general, (a Ϫ b)2 a2 Ϫ b2. Dis-49. a b ϩ c 2 2 2 50. 9x Ϫ 4 2 cuss possible conditions on a and b that would make this a valid equation.51. 4x ϩ 9 2 52. 16x2 Ϫ 25 89. To show that 12 is an irrational number, explain how the53. 6x2 ϩ 48x ϩ 72 54. 3z2 Ϫ 28z ϩ 48 assumption that 12 is rational leads to a contradiction of Theo-55. 2x4 Ϫ 24x3 ϩ 40x2 56. 16x2y Ϫ 8xy ϩ y rem 1, the fundamental theorem of arithmetic, by the following steps:57. 6m2 Ϫ mn Ϫ 12n2 58. 4u3v Ϫ uv3 (A) Suppose that 12 ϭ aրb, where a and b are positive inte-59. 3m3 Ϫ 6m2 ϩ 15m 60. 2x3 Ϫ 2x2 ϩ 8x gers, b 0. Explain why a2 ϭ 2b2. (B) Explain why the prime number 2 appears an even number61. m3 ϩ n3 62. 8x3 Ϫ 125 of times (possibly 0 times) as a factor in the prime factor- ization of a2. (C) Explain why the prime number 2 appears an odd number ofProblems 63–68 are calculus-related. Perform the indicated times as a factor in the prime factorization of 2b2.operations and simplify. (D) Explain why parts (B) and (C) contradict the fundamental63. 3(x ϩ h) Ϫ 7 Ϫ (3x Ϫ 7) theorem of arithmetic.64. (x ϩ h)2 Ϫ x2
• SECTION R–3 Polynomials: Basic Operations and Factoring 3190. To show that 1n is an irrational number unless n is a perfect 0.3 centimeters thick, write an algebraic expression in terms of x square, explain how the assumption that 1n is rational leads to that represents the volume of the plastic used to construct the con- a contradiction of the fundamental theorem of arithmetic by the tainer. Simplify the expression. [Recall: The volume V of a sphere following steps: of radius r is given by V ϭ 4␲r3.] 3 (A) Assume that n is not a perfect square, that is, does not be- 96. PACKAGING A cubical container for shipping computer com- long to the sequence 1, 4, 9, 16, 25, . . . . Explain why some ponents is formed by coating a metal mold with polystyrene. If prime number p appears an odd number of times as a fac- the metal mold is a cube with sides x centimeters long and the tor in the prime factorization of n. polystyrene coating is 2 centimeters thick, write an algebraic (B) Suppose that 1n ϭ aրb, where a and b are positive inte- expression in terms of x that represents the volume of the poly- gers, b 0. Explain why a2 ϭ nb2. styrene used to construct the container. Simplify the expression. (C) Explain why the prime number p appears an even number [Recall: The volume V of a cube with sides of length t is given by of times (possibly 0 times) as a factor in the prime factor- V ϭ t3.] ization of a2. (D) Explain why the prime number p appears an odd number of 97. CONSTRUCTION A rectangular open-topped box is to be con- times as a factor in the prime factorization of nb2. structed out of 20-inch-square sheets of thin cardboard by cutting (E) Explain why parts (C) and (D) contradict the fundamental x-inch squares out of each corner and bending the sides up as indi- theorem of arithmetic. cated in the ﬁgure. Express each of the following quantities as a polynomial in both factored and expanded form.APPLICATIONS (A) The area of cardboard after the corners have been removed. (B) The volume of the box.91. GEOMETRY The width of a rectangle is 5 centimeters less thanits length. If x represents the length, write an algebraic expressionin terms of x that represents the perimeter of the rectangle. Simplifythe expression. 20 inches92. GEOMETRY The length of a rectangle is 8 meters more than its x xwidth. If x represents the width of the rectangle, write an algebraic x xexpression in terms of x that represents its area. Change the expres-sion to a form without parentheses.93. COIN PROBLEM A parking meter contains nickels, dimes, and 20 inchesquarters. There are 5 fewer dimes than nickels, and 2 more quartersthan dimes. If x represents the number of nickels, write an algebraicexpression in terms of x that represents the value of all the coins in x xthe meter in cents. Simplify the expression. x x94. COIN PROBLEM A vending machinecontains dimes and quarters only. Thereare 4 more dimes than quarters. If x repre-sents the number of quarters, write analgebraic expression in terms of x thatrepresents the value of all the coins in thevending machine in cents. Simplify theexpression.95. PACKAGING A spherical plastic con-tainer for designer wristwatches has aninner radius of x centimeters (see the ﬁgure). If the plastic shell is 98. CONSTRUCTION A rectangular open-topped box is to be con- structed out of 9- by 16-inch sheets of thin cardboard by cutting x-inch squares out of each corner and bending the sides up. Express each of the following quantities as a polynomial in both factored and expanded form. (A) The area of cardboard after the corners have been removed. 0.3 cm (B) The volume of the box. x cm Figure for 95
• 32 CHAPTER R BASIC ALGEBRAIC OPERATIONS R-4 Rational Expressions: Basic Operations Z Reducing to Lowest Terms Z Multiplication and Division Z Addition and Subtraction Z Compound Fractions A quotient of two algebraic expressions, division by 0 excluded, is called a fractional expression. If both the numerator and denominator of a fractional expression are polyno- mials, the fractional expression is called a rational expression. Some examples of rational expressions are the following (recall that a nonzero constant is a polynomial of degree 0): xϪ2 1 3 x2 ϩ 3x Ϫ 5 2x Ϫ 3x ϩ 5 2 x Ϫ1 4 x 1 In this section, we discuss basic operations on rational expressions, including multiplica- tion, division, addition, and subtraction. Since variables represent real numbers in the rational expressions we are going to con- sider, the properties of real number fractions summarized in Section R-1 play a central role in much of the work that we will do. Even though not always explicitly stated, we always assume that variables are restricted so that division by 0 is excluded. Z Reducing to Lowest Terms We start this discussion by restating the fundamental property of fractions (from Theorem 3 in Section R-1): Z FUNDAMENTAL PROPERTY OF FRACTIONS If a, b, and k are real numbers with b, k 0, then ka a 2ؒ3 3 (x ؊ 3)2 2 ϭ 2ؒ4 ‫؍‬ 4 (x ؊ 3)x ‫؍‬ x kb b x 0, x 3 Using this property from left to right to eliminate all common factors from the numer- ator and the denominator of a given fraction is referred to as reducing a fraction to lowest terms. We are actually dividing the numerator and denominator by the same nonzero com- mon factor. Using the property from right to left—that is, multiplying the numerator and the denom- inator by the same nonzero factor—is referred to as raising a fraction to higher terms. We will use the property in both directions in the material that follows. We say that a rational expression is reduced to lowest terms if the numerator and denominator do not have any factors in common. Unless stated to the contrary, factors will be relative to the integers.
• SECTION R–4 Rational Expressions: Basic Operations 33EXAMPLE 1 Reducing Rational Expressions Reduce each rational expression to lowest terms. x2 Ϫ 6x ϩ 9 x3 Ϫ 1 (A) (B) x2 Ϫ 9 x2 Ϫ 1 SOLUTIONS x2 Ϫ 6x ϩ 9 (x Ϫ 3)2 Factor numerator and denominator completely. Divide numerator (A) ϭ and denominator by (x Ϫ 3); this is a valid operation as long as x2 Ϫ 9 (x Ϫ 3)(x ϩ 3) x 3. xϪ3 ϭ xϩ3 1 x3 Ϫ 1 (x Ϫ 1)(x2 ϩ x ϩ 1) Dividing numerator and denominator by (x ؊ 1) can be (B) 2 ϭ indicated by drawing lines through both (x ؊ 1)’s and x Ϫ1 (x Ϫ 1)(x ϩ 1) writing the resulting quotients, 1’s. 1 x2 ϩ x ϩ 1 ϭ x ؊1 and x 1 xϩ1MATCHED PROBLEM 1 Reduce each rational expression to lowest terms. 6x2 ϩ x Ϫ 2 x4 Ϫ 8x (A) (B) 2x2 ϩ x Ϫ 1 3x Ϫ 2x2 Ϫ 8x 3 ZZZ CAUTION Z Z Z Remember to always factor the numerator and denominator ﬁrst, then divide out any common factors. Do not indiscriminately eliminate terms that appear in both the numerator and the denominator. For example, 1 2x3 ϩ y2 2x3 ϩ y2 y2 y2 1 2x3 ϩ y2 2x3 ϩ 1 y2 Since the term y2 is not a factor of the numerator, it cannot be eliminated. In fact, (2x3 ϩ y2)րy2 is already reduced to lowest terms. Z Multiplication and Division Since we are restricting variable replacements to real numbers, multiplication and division of rational expressions follow the rules for multiplying and dividing real number fractions (Theorem 3 in Section R-1). Z MULTIPLICATION AND DIVISION If a, b, c, and d are real numbers with b, d 0, then: a c ac 2 x 2x 1. ؒ ϭ ؒ 3 x؊1 ‫؍‬ 3(x ؊ 1) b d bd a c a d 2 x 2 x؊1 2. Ϭ ϭ ؒ c 0 ، x؊1 ‫ؒ ؍‬ b d b c 3 3 x
• 34 CHAPTER R BASIC ALGEBRAIC OPERATIONS EXAMPLE 2 Multiplying and Dividing Rational Expressions Perform the indicated operations and reduce to lowest terms. 10x3y x2 Ϫ 9 4 Ϫ 2x (A) ؒ 2 (B) Ϭ (x Ϫ 2) 3xy ϩ 9y 4x Ϫ 12x 4 2x3 Ϫ 2x2y ϩ 2xy2 x3 ϩ y3 (C) Ϭ 2 x y Ϫ xy 3 3 x ϩ 2xy ϩ y2 5x2 1ؒ1 Factor numerators and denominators; 10x3y x2 Ϫ 9 10x3y (x Ϫ 3)(x ϩ 3) then divide any numerator and any SOLUTIONS (A) ؒ 2 ϭ ؒ denominator with a like common 3xy ϩ 9y 4x Ϫ 12x 3y(x ϩ 3) 4x(x Ϫ 3) factor. 3ؒ1 2ؒ1 2 5x ϭ 6 1 4 Ϫ 2x 2(2 Ϫ x) 1 x؊2 (B) Ϭ (x Ϫ 2) ϭ ؒ x ؊ 2 is the same as . 4 4 xϪ2 1 2 Ϫ1 2Ϫx Ϫ(x Ϫ 2) b ؊ a ‫(؊ ؍‬a ؊ b), a useful change ϭ ϭ in some problems. 2(x Ϫ 2) 2(x Ϫ 2) 1 1 ϭϪ 2 2x3 Ϫ 2x2y ϩ 2xy2 x3 ϩ y3 a c a d (C) Ϭ 2 ، ‫ؒ ؍‬ x3y Ϫ xy3 x ϩ 2xy ϩ y2 b d b c 2 1 1 2x(x2 Ϫ xy ϩ y2) (x ϩ y)2 ϭ ؒ Divide out common factors. xy(x ϩ y)(x Ϫ y) (x ϩ y)(x2 Ϫ xy ϩ y2) y 1 1 1 2 ϭ y(x Ϫ y) MATCHED PROBLEM 2 Perform the indicated operations and reduce to lowest terms. 12x2y3 y2 ϩ 6y ϩ 9 x2 Ϫ 16 (A) ؒ (B) (4 Ϫ x) Ϭ 2xy2 ϩ 6xy 3y3 ϩ 9y2 5 m3 ϩ n3 m3n Ϫ m2n2 ϩ mn3 (C) Ϭ 2m2 ϩ mn Ϫ n2 2m3n2 Ϫ m2n3 Z Addition and Subtraction Again, because we are restricting variable replacements to real numbers, addition and sub- traction of rational expressions follow the rules for adding and subtracting real number frac- tions (Theorem 3 in Section R-1).
• SECTION R–4 Rational Expressions: Basic Operations 35 Z ADDITION AND SUBTRACTION For a, b, and c real numbers with b 0: a c aϩc x 2 x؉2 1. ϩ ϭ x؊3 ؉ x؊3 ‫؍‬ x؊3 b b b a c aϪc x x؊4 x ؊ (x ؊ 4) 2. Ϫ ϭ 2xy 2 ؊ 2xy 2 ‫؍‬ 2xy 2 b b b So we add rational expressions with the same denominators by adding or subtracting their numerators and placing the result over the common denominator. If the denominators are not the same, we raise the fractions to higher terms, using the fundamental property of fractions to obtain common denominators, and then proceed as described. Even though any common denominator will do, our work will be simpliﬁed if the least common denominator (LCD) is used. Often, the LCD is obvious, but if it is not, the steps in the box describe how to ﬁnd it. Z THE LEAST COMMON DENOMINATOR (LCD) The LCD of two or more rational expressions is found as follows: 1. Factor each denominator completely. 2. Identify each different prime factor from all the denominators. 3. Form a product using each different factor to the highest power that occurs in any one denominator. This product is the LCD.EXAMPLE 3 Adding and Subtracting Rational Expressions Combine into a single fraction and reduce to lowest terms. 3 5 11 4 5x xϩ3 xϩ2 5 (A) ϩ Ϫ (B) Ϫ 2ϩ1 (C) Ϫ 2 Ϫ 10 6 45 9x 6y x Ϫ 6x ϩ 9 2 x Ϫ9 3Ϫx SOLUTIONS (A) To ﬁnd the LCD, factor each denominator completely: 10 ϭ 2 ؒ 5 ͮ 6 ϭ 2 ؒ 3 LCD ϭ 2 ؒ 32 ؒ 5 ϭ 90 45 ϭ 32 ؒ 5 Now use the fundamental property of fractions to make each denominator 90: 3 5 11 9ؒ3 15 ؒ 5 2 ؒ 11 ϩ Ϫ ϭ ϩ Ϫ Multiply. 10 6 45 9 ؒ 10 15 ؒ 6 2 ؒ 45 27 75 22 ϭ ϩ Ϫ Combine into a 90 90 90 single fraction. 27 ϩ 75 Ϫ 22 80 8 ϭ ϭ ϭ 90 90 9
• 36 CHAPTER R BASIC ALGEBRAIC OPERATIONS (B) 9x ϭ 32x 6y2 ϭ 2 ؒ 3y2 ͮ LCD ϭ 2 ؒ 32xy2 ϭ 18xy2 4 5x 2y2 ؒ 4 3x ؒ 5x 18xy2 Ϫ 2ϩ1ϭ 2 Ϫ ϩ Multiply, combine. 9x 6y 2y ؒ 9x 3x ؒ 6y2 18xy2 8y2 Ϫ 15x2 ϩ 18xy2 ϭ 18xy2 xϩ3 xϩ2 5 xϩ3 xϩ2 5 (C) Ϫ 2 Ϫ ϭ Ϫ ϩ x2 Ϫ 6x ϩ 9 x Ϫ9 3Ϫx (x Ϫ 3)2 (x Ϫ 3)(x ϩ 3) xϪ3 5 5 5 We have again used the fact Note: Ϫ ϭϪ ϭ that a ؊ b ‫(؊ ؍‬b ؊ a). 3Ϫx Ϫ(x Ϫ 3) xϪ3 The LCD ϭ (x Ϫ 3)2(x ϩ 3). (x ϩ 3)2 (x Ϫ 3)(x ϩ 2) 5(x Ϫ 3)(x ϩ 3) Ϫ ϩ Expand numerators. (x Ϫ 3) (x ϩ 3) 2 (x Ϫ 3) (x ϩ 3) 2 (x Ϫ 3)2(x ϩ 3) (x2 ϩ 6x ϩ 9) Ϫ (x2 Ϫ x Ϫ 6) ϩ 5(x2 Ϫ 9) Be careful of sign ϭ (x Ϫ 3)2(x ϩ 3) errors here. x2 ϩ 6x ϩ 9 Ϫ x2 ϩ x ϩ 6 ϩ 5x2 Ϫ 45 ϭ Combine like terms. (x Ϫ 3)2(x ϩ 3) 5x2 ϩ 7x Ϫ 30 ϭ (x Ϫ 3)2(x ϩ 3) MATCHED PROBLEM 3 Combine into a single fraction and reduce to lowest terms. 5 1 6 1 2x ϩ 1 3 (A) Ϫ ϩ (B) 2 Ϫ 3 ϩ 28 10 35 4x 3x 12x yϪ3 yϩ2 2 (C) Ϫ 2 Ϫ y2 Ϫ 4 y Ϫ 4y ϩ 4 2ϪyZZZ EXPLORE-DISCUSS 1 What is the result of entering 16 Ϭ 4 Ϭ 2 on a calculator? What is the difference between 16 Ϭ (4 Ϭ 2) and (16 Ϭ 4) Ϭ 2? How could you use fraction bars to distinguish between these two cases when 16 writing 4 ? 2 Z Compound Fractions A fractional expression with fractions in its numerator, denominator, or both is called a compound fraction. It is often necessary to represent a compound fraction as a simple fraction—that is (in all cases we will consider), as the quotient of two polynomials. The process does not involve any new concepts. It is a matter of applying old concepts and processes in the right sequence. We will illustrate two approaches to the problem, each with its own merits, depending on the particular problem under consideration.
• SECTION R–4 Rational Expressions: Basic Operations 37 EXAMPLE 4 Simplifying Compound Fractions Express as a simple fraction reduced to lowest terms: 2 Ϫ1 x 4 Ϫ1 x2 SOLUTION Method 1. Multiply the numerator and denominator by the LCD of all fractions in the numerator and denominator—in this case, x2. (We are multiplying by 1 ϭ x2րx2.) 2 2 x2a Ϫ 1b x2 Ϫ x2 1 x x 2x Ϫ x2 x(2 Ϫ x) ϭ ϭ 2 ϭ 4 4 4Ϫx (2 ϩ x)(2 Ϫ x) x 2a Ϫ 1b x2 Ϫ x2 1 x2 x2 x ϭ 2ϩx Method 2. Write the numerator and denominator as single fractions. Then treat as a quotient. 2 2Ϫx Ϫ1 1 x x x 2Ϫx 4 Ϫ x2 2Ϫx x2 ϭ ϭ Ϭ ϭ ؒ 4 4 Ϫ x2 x x2 x (2 Ϫ x)(2 ϩ x) 2 Ϫ 1 1 1 2 x x x ϭ 2ϩx MATCHED PROBLEM 4 Express as a simple fraction reduced to lowest terms. Use the two methods described in Example 4. 1 1ϩ x 1 xϪ x ANSWERS TO MATCHED PROBLEMS 3x ϩ 2 x2 ϩ 2x ϩ 4 Ϫ5 1. (A) (B) 2. (A) 2x (B) (C) mn xϩ1 3x ϩ 4 xϩ4 1 3x Ϫ 5x Ϫ 4 2 2y Ϫ 9y Ϫ 6 2 1 3. (A) (B) (C) 4. 4 12x 3 ( y Ϫ 2) ( y ϩ 2) 2 xϪ1 R-4 ExercisesIn Problems 1–10, reduce each rational expression to lowest terms. xϩ1 x2 Ϫ 2x Ϫ 24 x2 Ϫ 9 5. 6. 7. 17 91 360 63 x ϩ 3x ϩ 2 2 xϪ6 x ϩ 3x Ϫ 18 21. 2. 3. 4. 85 26 288 105
• 38 CHAPTER R BASIC ALGEBRAIC OPERATIONS x2 ϩ 9x ϩ 20 3x2y3 2a2b4c6 Ϫ2x(x ϩ 4)3 Ϫ 3(3 Ϫ x2)(x ϩ 4)28. 9. 10. 41. x2 Ϫ 16 x4y 6a5b3c (x ϩ 4)6 3x2(x ϩ 1)3 Ϫ 3(x3 ϩ 4)(x ϩ 1)2In Problems 11–36, perform the indicated operations and reduce 42.answers to lowest terms. Represent any compound fractions as (x ϩ 1)6simple fractions reduced to lowest terms. In Problems 43–54, perform the indicated operations and reduce 5 11 7 19 answers to lowest terms. Represent any compound fractions as11. ϩ 12. ϩ 6 15 10 25 simple fractions reduced to lowest terms. 1 1 9 8 y 2 113. Ϫ 14. Ϫ 43. Ϫ ϩ 2 8 9 8 9 y Ϫ 2y Ϫ 8 2 y2 Ϫ 5y ϩ 4 y ϩyϪ2 1 1 m n x xϪ8 xϩ415. Ϫ 16. Ϫ 44. ϩ ϩ n m n m x2 Ϫ 9x ϩ 18 xϪ6 xϪ3 5 3 10 5 16 Ϫ m2 mϪ117. Ϭ 18. Ϭ 45. ؒ 12 4 3 2 m ϩ 3m Ϫ 4 m Ϫ 4 2 25 5 4 25 5 4 xϩ1 x2 Ϫ 2x ϩ 119. a Ϭ bؒ 20. Ϭa ؒ b 46. ؒ 8 16 15 8 16 15 x(1 Ϫ x) x2 Ϫ 1 b2 b a b2 b a xϩ7 yϩ921. a Ϭ 2b ؒ 22. Ϭa 2ؒ b 47. ϩ 2a a 3b 2a a 3b ax Ϫ bx by Ϫ ay x2 Ϫ 1 xϩ1 x2 Ϫ 9 xϪ3 cϩ2 cϪ2 c23. Ϭ 2 24. Ϭ 48. Ϫ ϩ xϩ2 x Ϫ4 x Ϫ1 2 xϪ1 5c Ϫ 5 3c Ϫ 3 1Ϫc 1 1 1 1 1 1 x2 Ϫ 16 x2 Ϫ 13x ϩ 3625. ϩ ϩ 26. ϩ ϩ 49. Ϭ c b a bc ac ab 2x ϩ 10x ϩ 8 2 x3 ϩ 1 2a Ϫ b 2a ϩ 3b x3 Ϫ y3 y x2 ϩ xy ϩ y227. 2 Ϫ 2 50. a ؒ bϬ a Ϫb 2 a ϩ 2ab ϩ b2 y3 xϪy y2 xϩ2 xϪ2 x 1 428. Ϫ 51. a Ϫ bϬ x2 Ϫ 1 (x Ϫ 1)2 x2 Ϫ 16 xϩ4 xϩ4 mϪ2 xϩ1 3 1 xϩ429. m ϩ 2 Ϫ 30. ϩx 52. a Ϫ bϬ mϪ1 xϪ1 xϪ2 xϩ1 xϪ2 3 2 1 2 2 15 x y31. Ϫ 32. Ϫ 1ϩϪ 2 Ϫ2ϩ xϪ2 2Ϫx aϪ3 3Ϫa x x y x 53. 54. 4 5 x y 3 2 4y 1ϩ Ϫ 2 Ϫ33. ϩ Ϫ 2 x x y x yϩ2 yϪ2 y Ϫ4 4x 3 2 Problems 55–58 are calculus-related. Perform the indicated34. ϩ Ϫ operations and reduce answers to lowest terms. Represent any x Ϫy 2 2 xϩy xϪy compound fractions as simple fractions reduced to lowest terms. x2 4 Ϫ1 Ϫx 1 1 1 1 y2 x Ϫ Ϫ 235. 36. xϩh x (x ϩ h)2 x x 2 55. 56. ϩ1 Ϫ1 h h y x (x ϩ h)2 x2 2x ϩ 2h ϩ 3 2x ϩ 3Problems 37–42 are calculus-related. Reduce each fraction to Ϫ Ϫ xϩhϩ2 xϩ2 xϩh xlowest terms. 57. 58. h h 6x3(x2 ϩ 2)2 Ϫ 2x(x2 ϩ 2)337. x4 In Problems 59–62, perform the indicated operations and reduce 4x (x ϩ 3) Ϫ 3x2(x2 ϩ 3)2 4 2 answers to lowest terms. Represent any compound fractions as38. x6 simple fractions reduced to lowest terms. 2x(1 Ϫ 3x)3 ϩ 9x2(1 Ϫ 3x)2 y2 s239. yϪ Ϫs (1 Ϫ 3x) 6 yϪx sϪt 59. 60. 2 x2 t 2x(2x ϩ 3)4 Ϫ 8x2(2x ϩ 3)3 1ϩ 2 ϩt40. y Ϫ x2 sϪt (2x ϩ 3)8
• Review 39 1 1 Discuss possible conditions of a and b that would make this a61. 2 Ϫ 62. 1 Ϫ 2 1 valid equation. 1Ϫ 1Ϫ aϩ2 1 64. Show by example that, in general, 1Ϫ x a2 ϩ b263. Show by example that, in general, a ϩ b (assume a Ϫb) aϩb aϩb Discuss possible conditions of a and b that would make this a aϩ1 (assume b 0) b valid equation. CHAPTER R ReviewR-1 Algebra and Real Numbers a c aϪc a c ad ϩ bc 6. Ϫ ϭ 7. ϩ ϭA real number is any number that has a decimal representation. b b b b d bdThere is a one-to-one correspondence between the set of real num-bers and the set of points on a line. Important subsets of the real R-2 Exponents and Radicalsnumbers include the natural numbers, integers, and rational The notation an, in which the exponent n is an integer, is deﬁned asnumbers. A rational number can be written in the form aրb, where follows. For n a positive integer and a a real number:a and b are integers and b 0. A real number can be approximatedto any desired precision by rational numbers. Consequently, arith- an ϭ a ؒ a ؒ . . . ؒ a (n factors of a)metic operations on rational numbers can be extended to operations 1on real numbers. These operations satisfy basic real number prop- aϪn ϭ (a 0)erties, including associative properties: x ϩ ( y ϩ z) ϭ an(x ϩ y) ϩ z and x( yz) ϭ (xy)z; commutative properties: a0 ϭ 1 (a 0)x ϩ y ϭ y ϩ x and xy ϭ yx; identities: 0 ϩ x ϭ x ϩ 0 ϭ x and(1)x ϭ x(1) ϭ x; inverses: Ϫx is the additive inverse of x and, if Properties of integer exponents (division by 0 excluded):x 0, xϪ1 is the multiplicative inverse of x; and distributive 1. aman ϭ amϩn 2. (an)m ϭ amnproperty: x( y ϩ z) ϭ xy ϩ xz. Subtraction is deﬁned bya Ϫ b ϭ a ϩ (Ϫb) and division by aրb ϭ abϪ1. Division by 0 is a m am 3. (ab)m ϭ ambm 4. a b ϭ mnever allowed. Additional properties include properties of negatives: b b am 11. Ϫ(Ϫa) ϭ a 5. ϭ amϪn ϭ nϪm an a2. (Ϫa)b ϭ Ϫ(ab) ϭ a(Ϫb) ϭ Ϫab Any positive number written in decimal form can be ex-3. (Ϫa)(Ϫb) ϭ ab pressed in scientiﬁc notation, that is, in the form a ϫ 10n 1 Յ a 6 10, n an integer, a in decimal form.4. (Ϫ1)a ϭ Ϫa For n a natural number, a and b real numbers: a is an nth root Ϫa a a of b if an ϭ b. The number of real nth roots of a real number b is5. ϭϪ ϭ b 0 either 0, 1, or 2, depending on whether b is positive or negative, and b b Ϫb whether n is even or odd. The principal nth root of b, denoted by n Ϫa Ϫa a a b1/n or 1b, is the real nth root of b if there is only one, and the pos-6. ϭϪ ϭϪ ϭ b 0 n Ϫb b Ϫb b itive nth root of b if there are two real nth roots. In the notation 1b, the symbol 1 is called a radical, n is called the index, and b is thezero properties: radicand. If n ϭ 2 we write 1b in place of 1b. 21. a ؒ 0 ϭ 0 We extend exponent notation so that exponents can be rational numbers, not just integers, as follows. For m and n natu-2. ab ϭ 0 if and only if aϭ0 or bϭ0 or both. ral numbers and b any real number (except b cant be negativeand fraction properties (division by 0 excluded): when n is even), 11. a ϭ c if and only if ad ϭ bc bmրn ϭ (b1րn)m and bϪmրn ϭ b d bm ր n ka a a c ac Rational exponent/radical property:2. ϭ 3. ؒ ϭ (b1րn)m ϭ (bm)1րn and n n kb b b d bd (1b)m ϭ 2bm a c a d a c aϩc4. Ϭ ϭ ؒ 5. ϩ ϭ b d b c b b b