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Barnett/Ziegler/Byleen Precalculus: A Graphing Approach Appendix A Basic Algebra Review Copyright © 2000 by the McGraw-Hill Companies, Inc. The Set of Real Numbers N Natural Numbers 1, 2, 3, . . . Z Integers . . . , –2, –1, 0, 1, 2, . . . Rational Numbers __ –3 2 –4, 0, 8, 5 , 3 , 3.14, –5.2727 Q I R Irrational Numbers Real Numbers 2, 3 –7, 0, 3 7 , 1.414213 . . . –2 – 5 , 3 , 3.14, 0.333 , Copyright © 2000 by the McGraw-Hill Companies, Inc. A-1-113 Subsets of the Set of Real Numbers Natural numbers (N) Zero Integers (Z) Rational numbers (Q) Negatives of natural numbers Noninteger ratios of integers N Z Q Real numbers (R) Irrational numbers (I) R Copyright © 2000 by the McGraw-Hill Companies, Inc. A-1-114 Basic Real Number Properties Let R be the set of real numbers and let x, y, and z be arbitrary elements of R. Addition Properties Closure: x + y is a unique element in R. Associative: Commutative: Identity: Inverse: (x + y ) + z = x + (y + z ) x +y =y +x 0 +x = x + 0 =x x + (–x ) = (–x ) + x = 0 Copyright © 2000 by the McGraw-Hill Companies, Inc. A-1-115(a) Basic Real Number Properties Multiplication Properties Closure: xy is a unique element in R . Associative: (xy)z = x(yz) Commutative: xy = yx Identity: (1) x = x (1) = x Inverse: X 1 1 = x x x =1x 0 Combined Property Distributive: x (y + z ) = xy + xz ( x + y ) z = xz + yz Copyright © 2000 by the McGraw-Hill Companies, Inc. A-1-115(b) Foil Method F (2x – 1)(3x + 2) O I L First Outer Inner Last Product Product Product Product = – – 6x2 + 4x 3x 2 Special Products 2 1. (a – b)(a + b) = a – 2 b 2. (a + b)2 = a2 + 2ab + b2 3. (a – b)2 = a2 – 2ab + b2 Copyright © 2000 by the McGraw-Hill Companies, Inc. A-2-116 Special Factoring Formulas 1. u 2 + 2 uv + v 2 = ( u + v )2 Perfect Square 2. u 2 – 2uv + v 2 = (u – v) 2 Perfect Square 3. u 2 – v 2 = (u – v)(u + v) Difference of Squares 4. u 3 – v 3 = (u – v)(u 2 + uv + v 2) Difference of Cubes 5. u 3 + v 3 = (u + v)(u2 – uv + v 2) Sum of Cubes Copyright © 2000 by the McGraw-Hill Companies, Inc. A-3-117 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. Copyright © 2000 by the McGraw-Hill Companies, Inc. A-4-118 Definition of an 1. For n a positive integer: an = a · a · … · a n factors of a Exponent Properties 1. a m a n = a m + n 2. ( a n) m = a mn 3. (ab)m = a m bm 2. For n = 0 , a0 = 1 a 0 00 is not defined am am 4. = m b b b0 3. For n a negative integer, 1 an = –n a a 0 am 1 5. n = a m–n = n–m a a Copyright © 2000 by the McGraw-Hill Companies, Inc. a0 A-5-119 Definition of b1/n For n a natural number and b a real number, b1/n is the principal nth root of b defined as follows: 1. If n is even and b is positive, then b1/n represents the positive nth root of b. 2. If n is even and b is negative, then b1/n does not represent a real number. 3. If n is odd, then b1/n represents the real nth root of b (there is only one). 4. 01/n = 0 Rational Exponents b = ( b m )1/n Copyright © 2000 by the McGraw-Hill Companies, Inc. For m and n natural numbers and b any real number (except b cannot be negative when n is even): ( b1/ n ) m m/n A-6-120 n b , nth-Root Radical For n a natural number greater than 1 and b a real number, we define be the principal nth root of b; that is, n If n = 2, we write b in place of 2 n b to b = b1/n b. Rational Exponent/ Radical Conversions For m and n positive integers (n > 1), and b not negative when n is even, 1. 2. n n (bm)1/n = n bm bm/n = n m 1/n m (b ) = ( b) xn = x xy = n x n x nx 3. y =n y n y Properties of Radicals Copyright © 2000 by the McGraw-Hill Companies, Inc. A-7-121 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, x5 violates this condition.) 2. No power of the radicand and the index of the radical have a common factor other than 1. (For example, 6 x4 violates this condition.) 3. No radical appears in a denominator. y (For example, violates this condition.) x 4. No fraction appears within a radical. 3 (For example, 5 violates this condition.) Copyright © 2000 by the McGraw-Hill Companies, Inc. A-7-122 Interval Notation Interval Notation Inequality Notation [a, b] a x b [a, b) a x<b (a, b] a<x b (a, b) a<x<b Line Graph Type [ ] x Closed [ ) x Half-open ( ] x Half-open ( ) x Open a a a a b b b b Copyright © 2000 by the McGraw-Hill Companies, Inc. A-8-123(a) Interval Notation Interval Notation [b , ) ( b, ) ( –, a] ( –, a) Inequality Notation Line Graph Type x b [ b x x x> b ( b x a x< a Closed Open ] a x ) a x Copyright © 2000 by the McGraw-Hill Companies, Inc. Closed Open A-8-123(b) Inequality Properties For a, b, and c any real numbers: 1. If a < b and b < c, then a < c. Transitive Property 2. If a < b, then a + c < b + c. Addition Property 3. If a < b, then a – c < b – c. Subtraction Property 5. If a < b and c is negative, then ca > cb. Multiplication Property (Note difference between 4 and 5.) Division Property (Note difference between 6 and 7.) 4. If a < b and c is positive, then ca < cb. a b 6. If a < b and c is positive, then c < . c a b 7. If a < b and c is negative, then c > c . Copyright © 2000 by the McGraw-Hill Companies, Inc. A-8-124 Partial Fraction Decomposition Any proper fraction P(x)/D(x) reduced to lowest terms can be decomposed in the sum of partial fractions as follows: 1. If D(x) has a nonrepeating linear factor of the form ax + b, then the partial fraction decomposition of P(x)/D(x) contains a term of the form A ax + b A a constant 2. If D(x) has a k-repeating linear factor of the form (ax + b)k, then the partial fraction decomposition of P(x)/D(x) contains k terms of the form A1 A2 Ak A1 , A2 , …, Ak constants ax + b + (ax + b)2 + … + (ax + b)k 3. If D(x) has a nonrepeating quadratic factor of the form ax2 + bx + c, which is prime relative to the real numbers, then the partial fraction decomposition of P(x)/D(x) contains a term of the form Ax + B ax 2 + bx + c A, B constants 4. If D(x) has a k-repeating quadratic factor of the form (ax2 + bx + c)k, where ax2 + bx + c is prime relative to the real numbers, then the partial fraction decomposition of P(x)/D(x) contains k terms of the form A1 x + B1 A2 x + B2 Ak x + Bk + +…+ (ax2 + bx + c)k ax2 + bx + c (ax2 +bx + c)2 A1 , …, Ak , B1 , …, Bk constants Copyright © 2000 by the McGraw-Hill Companies, Inc. B-1-125 Significant Digits If a number x is written in scientific notation as x = a 10n 1 a < 10 , n an integer then the number of significant digits in x is the number of digits in a. The number of significant digits in a number with no decimal point if found by counting the digits from left to right, starting with the first digit and ending with the last nonzero digit. The number of significant digits in a number containing a decimal point is found by counting the digits from left to right, starting with the first nonzero digit and ending with the last digit. Rounding Calculated Values The result of a calculation is rounded to the same number of significant digits as the number used in the calculation that has the least number of significant digits. Copyright © 2000 by the McGraw-Hill Companies, Inc. C-1-126