Copyright © 2000 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen Precalculus: A Graphing Approach Appendix A Basic Algebra Review

Copyright © 2000 by the McGraw-Hill Companies, Inc. NNatural Numbers1, 2, 3,... ZIntegers..., –2, –1, 0, 1, 2,... QRational Numbers–4, 0, 8, –3 5, 2 3, 3.14, – __ IIrrational Numbers2,  3 7, RReal Numbers–7, 0, 3 5, –2 3, 3.14, –,  The Set of Real Numbers A-1-113

Copyright © 2000 by the McGraw-Hill Companies, Inc. Subsets of the Set of Real Numbers Natural numbers (N) Negatives of natural numbers Zero Integers (Z) Noninteger ratios of integers Rational numbers (Q) Irrational numbers (I) Real numbers (R) N  Z  Q  R A-1-114

Copyright © 2000 by the McGraw-Hill Companies, Inc. 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:(x +y ) + z = x + ( y + z ) Commutative:x +y =y +x Identity:0 +x =x + 0 = x Inverse:x + (– x ) = (– x ) + x = 0 A-1-115(a)

Copyright © 2000 by the McGraw-Hill Companies, Inc. Basic Real Number Properties Multiplication Properties Closure: xy is a unique element in R. Associative: Commutative:xy =yx Identity:(1) x = x (1) = x Inverse: X       1 x =       1 x x = 1x  0 Combined Property Distributive:x ( y + z) = xy + xz ( x + y ) z = + yz (xy)z = x(yz) A-1-115(b)

Copyright © 2000 by the McGraw-Hill Companies, Inc. Foil Method FOIL FirstOuterInnerLast Product  (2x – 1)(3x + 2) = 6x 2 +4x – 3x – 2 1.(a – b)( a +b ) = a 2 – b 2 2.(a +b) 2 =a 2 + 2ab +b 2 3.(a – b) 2 =a 2 – 2ab +b 2 A Special Products

Copyright © 2000 by the McGraw-Hill Companies, Inc. 1. Perfect Square 2.u 2 – 2 uv +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 )( u 2 –uv + v 2 ) Sum of Cubes u uv + v 2 = ( u +v ) 2 Special Factoring Formulas A-3-117

Copyright © 2000 by the McGraw-Hill Companies, Inc. 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. A-4-118

Copyright © 2000 by the McGraw-Hill Companies, Inc. 1.Forn a positive integer: a n =a ·a · … ·a n factors of a 2.Forn = 0, a 0 = 1a  is not defined 3.Forn a negative integer, a n = 1 a –n a  0 1.a m a n =a m + n 2. () a nm =a mn 3.(ab) m =a m b m 4.       a b m = a m b m b  0 5. a m a n =a m–n = 1 a n–m a  0 A Definition of a n Exponent Properties

Copyright © 2000 by the McGraw-Hill Companies, Inc. For n a natural number and b a real number, b 1/n is the principal nth root of b defined as follows: 1. If n is even and b is positive, then b 1/n represents the positive nth root of b. 2. If n is even and b is negative, then b 1/n does not represent a real number. 3. If n is odd, then b 1/n represents the real nth root of b (there is only one) /n = 0 For m and n natural numbers and b any real number (except b cannot be negative when n is even): A Definition of b 1/n Rational Exponents

Rational Exponent/ Radical Conversions Properties of Radicals A Copyright © 2000 by the McGraw-Hill Companies, Inc.

Simplified (Radical) Form A-7-122

Copyright © 2000 by the McGraw-Hill Companies, Inc. [a, b]a  x  b [] ab x Closed [a, b)a  x <b b [ a ) x Half-open (a, b]a <x  b ] a b x ( Half-open (a, b)a <x <b ab x () Open Interval Inequality Notation Notation Line Graph Type Interval Notation A-8-123(a)

Copyright © 2000 by the McGraw-Hill Companies, Inc. [b,)x  b b x [ Closed ( b,  )x >b b x ( Open ( –,–, a]x  a a x ] Closed (– ,, a)x <a a x ) Open Interval Inequality Notation Notation Line Graph Type  Interval Notation A-8-123(b)

Copyright © 2000 by the McGraw-Hill Companies, Inc. 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 4.If a < b and c is positive, then ca < cb. 5.If a < b and c is negative, then ca > cb.   Multiplication Property (Note difference between 4 and 5.) 6.If a < b and c is positive, then a c < b c. 7.If a < b and c is negative, then a c > b c.   Division Property (Note difference between 6 and 7.) For a, b, and c any real numbers: Inequality Properties A-8-124

Copyright © 2000 by the McGraw-Hill Companies, Inc. 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 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 3. If D(x) has a nonrepeating quadratic factor of the form ax 2 + 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 4. If D(x) has a k-repeating quadratic factor of the form (ax 2 + bx + c) k, where ax 2 + 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 A ax + b A 1 + b + A 2 ( + b ) 2 + … + A k ( ax + b ) k A 1, A 2, …, A k constants Ax + B ax 2 + bx + c A, B constants A 1 x + B 1 ax 2 + bx + c + A 2 x + B 2 ( ax 2 + bx + c ) 2 + … + A k x + B k ( ax 2 + bx + c ) k A 1, …, A k, B 1 B k constants Partial Fraction Decomposition B-1-125

Copyright © 2000 by the McGraw-Hill Companies, Inc. Significant Digits If a number x is written in scientific notation as x = a  10 n 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. C-1-126