P. Real Numbers ad Their Properties Natural Numbers 1,,3. Whole Numbers 0, 1,,... Itegers..., -1, 0, 1,... Real Numbers Ratioal umbers (p/q) Where p & q are itegers, q 0 Irratioal umbers o-termiatig ad o-repeatig) Property Commutative Associative Distributive Eistece of the additive idetity Eistece of the multiplicative idetity Eistece of a additive iverse Eistece of the multiplicative iverse Descriptio Order does t matter Groupig does t matter There eists a umber (zero) whose sum with ay umber is the umber. There is a umber (1) whose product with ay umber is that umber. There is a secod umber that whe added to the first gives zero. Algebraic descr. Properties of Negatio Let a ad b be real umbers, variables or algebraic epressios. 1. ( 1)a = a. a ( )= a 3. ( a)b = ( ab)= a b 4. ( a) b ( )= ab 5. ( a + b)= ( a)+ b ( ) ( ) 1
Properties of Fractios 1. Equivalet fractios: a b = c if ad oly if ad = bc. d. Rules of sigs: a b = a b = a ad a b b = a b 3. Geerate equivalet fractios: a b = ac bc, c 0 4. Add or subtract with like deomiators: a b ± c b = a ± c b 5. Add or subtract with ulike deomiators: a b ± c d 6. Multiply fractios: a b c d = ac bd 7. Divide fractios: a b c d = a b d c = ad bc Eamples: 3 + 3 = 3 + 1 5 = 3 5 3 = = ad ± bc bd
P.3 The Real Number Lie ad Order Defiitio of Order o the Real Number Lie a < b ad b > a, for a ad b real umbers, we say that a is less tha b or b is greater tha a. Sets ad Itervals A = {1,, 3, 4, 5, 6, 7} A = { is a iteger ad 0 < < 8} U = Uio = Itersectio Defiitio of Absolute Value Let be a real umber. The absolute value of deoted by, is if 0 = if < 0 = 10 Properties of Absolute Values Let a ad b be real umbers. The the followig properties are true. 1. a 0. -a = a 3. ab = a b 4. a b = a b,b 0 Distace Betwee Two Numbers Let a ad b be real umbers. The distace betwee a ad b is give by Distace = b a = a b distace betwee 3 ad 6 3 4 5 6 7 3
P.4 Iteger Epoets Properties of Epoets Property a m a = a m + Eample a a p p q = a q ( a m ) = a m ( ab) m = a m b m q a a = b b 1 a = a 0 a = 1 q q a = a = a y 4 z 3 = Scietific Notatio 7 = 7 10 0 0.003 = 3. 10-3 70 = 7. 10 67,000,000.005 = 4
Iterest Formulas t r Simple Iterest -- A= P( 1+ rt) Compoud Iterest -- A= P 1+ How much savigs would you have if you ivested $4000 for 5 years at 4%? a) simple iterest b) compouded daily 5
P.5 Radicals ad Ratioal Epoets I the epressio, is called a radical, the umber is called the ide, ad is called the radicad. If o ide is give, it is assumed to be. b = b 1/ Properties of radicals Property ( ) = For odd For eve y = = y y m m ( ) = m = = m y = Eample Eamples: Simplify the followig: 18 5 3 4 7 y 6 6
Ratioalizig a deomiator 7 4 Ratioalize the umerator + + 3 Additio ad subtractio of radical epressios 5 3 4 3+ 6 3 3 8+ 5 18 a /5 a 1/5 = 3 Covert to solve: 5 4 5 = 7
P.6 Algebraic Epressios A polyomial i is a epressio of the form a +L+ a + a 1 + a 0 Sums ad Differeces of Polyomials ( 3 ) ( 3 ) 3 y+ 4 y y+ 3 + y+ 3 y 4 ( ) ( ) 3 a ab b 7 a 5 ab+ 4 b Multiplyig Biomials FOIL method or distributive method 1 + 3 = 1 + 1 3 = ( )( ) ( )( ) ( )( ) or FOIL (first, outside, iside, last) ( ) ( 3) ( 1)( ) ( 1)( 3) + + + = ( + 3) ( 4) ( + 5)3 ( ) 8
Special Products a b = ( a + b )( a b ) Differece of squares a+ b = a + ab+ b ( ) Square of a Biomial ( ) a b = a ab+ b a b = a 3a b+ 3ab b 3 Cube of a Biomial ( ) 3 3 ( ) 3 3 a+ b = a + 3a b+ 3ab + b 3 A ope bo is made by cuttig squares from the corers of a piece of metal that measures 10 by 1 iches ad turig up the sides. The sides of the cut-out squares are all iches log, so the bo is iches tall. Fid the volume whe =. 9
P.7 Factorig Removig Commo Factors 6 5 3 1 3 + 6 + 9 a a a a Factorig Special Polyomial Forms a b = ( a+ b)( a b) Differece of squares a + ab+ b = ( a+ b) Perfect Squares Differece of cubes Sum of cubes ( ) a ab+ b = a b ( )( ) ( )( ) a b = a b a + ab+ b 3 3 a + b = a+ b a ab+ b 3 3 Factorig the Differece of Squares 4 9 ( 3) y Factorig Cubes 3 7 10
Factorig a Triomial 5 6 Factorig by groupig y + + 3y + 6 Factorig Guidelies a) If the polyomial has a greatest commo factor other tha 1, the factor out the greatest commo factor. b) If the polyomial has two terms, the see if is the differece of two squares, or the sum or differece of two cubes. Remember if it is the sum of two squares it will ot factor. c) If the polyomial has three terms, the it is either a perfect square triomial or if it is ot i which case you use oe of the trial ad error methods. d) If the polyomial has more tha three terms, the try to factor it by groupig. Fial check, look ad see if ay of the factors you have writte ca be factored further 11
P.8 Ratioal Epressios Domai of a epressio- All the values that are valid. The domai for all polyomials - Domai for ratioal epressio 4 Fid the domai 1 Domai for radicals Fid the domai 3 Simplifyig a Ratioal Epressio 4 3+ Multiplyig a Ratioal Epressio 3 1 + 3 1
Dividig a Ratioal Epressio y + y + 3 3 3 y y y Combiig Ratioal Epressios The LCD of two or more ratioal epressios is foud as follows: 1. Factor each deomiator completely.. Idetify each differet prime factor from all the deomiators. 3. Form a product usig each differet factor to the highest power that occurs i ay oe deomiator. This product is the LCD. 7 0 + 3 75 + 1 4 y 3 y+ 4 4 + 4 y y y y 13
Compoud Fractios 1 1 y 1 1 + y 3 4 14