Factoring Polynomials

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Factoring Polynomials"

Transcription

1 Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles rised to some non-negtive integer exponents. A polynomil with only one term is clled monomil, with two terms is clled binomil nd with three terms is clled trinomil. A polynomil in vrible x ( single-vrible polynomil) is of the form: p(x) = n x n + n x n + n x n x + x + 0 where ll of the coefficients ( n, n, n,... +,, 0 ) re rel numbers nd ll of the exponents (n, n, n,...,,, ) re non-negtive integers. The term 0 is clled the constnt term. The degree of term of polynomil is the sum of the exponents of the vribles of tht term. The degree of polynomil is the lrgest degree of ny of the individul terms of tht polynomil. The root of polynomil p(x) is number,, such tht p() = 0. The root or x-intercept or zero of the corresponding polynomil eqution p(x) = 0 is the number,, tht mkes p() = 0. If is root of polynomil p(x), then p(x) = (x )(q(x) for some polynomil q(x). The process of identifying some or ll such fctors (x ) is clled fctoring the polynomil. There re mny different types of fctoring techniques tht cn be pplied, none of which work for ll polynomils, so determining which fctoring method to be pplied requires some skill nd persistence. The Fundmentl Theorem of Algebr sttes: Every polynomil eqution hving complex coefficients nd degree n hs t lest one complex root. (On lighter note, the Frivolous Theorem of Arithmetic sttes: Almost ll nturl numbers re very, very, very lrge.) This cn be written symboliclly s: p(x) = n x n + n x n + n x n x + x + 0 = n (x z n ) (x z n ) (x z n )... (x z ) (x z ) n = n ( x z i ) (clled stndrd form) i= The theorem cn lso be tken to men tht ny polynomil with rel coefficients hs t lest one root, which cn be rel or complex. It cn be extended to imply tht ny polynomil with rel coefficients cn be cn be fctored into the product of liner or qudrtic roots (which is pplicble to secondry school mthemtics). For ll remining definitions nd exmples, we shll ssume tht ll coefficients of the polynomils nd the polynomil equtions (or functions) re rel numbers, unless otherwise stted. A trinomil which hs leding coefficient of is clled monic qudrtic trinomil (e.g. x + 7x +) nd those which hve the leding coefficient (i.e. in x + bx + c) not equl to is clled nonmonic qudrtic trinomil. This distinction usully requires different pproch when fctoring. The term of the polynomil (in one vrible) with the highest exponent is clled the leding term or the dominnt term. The polynomil lso derives is nme (in terms of its degree) from this leding term. Polynomils re clssified s liner (for x + b), qudrtic (x + bx + c), cubic (x + bx + cx + d), qurtic (x 4 + bx + cx + dx + e), etc., for leding term exponents of,,, 4, etc. A polynomil of the form (x ) n is sid be degenerte (or to hve degenerte roots) s ll roots re equl to.

2 A polynomil of degree n hs exctly n roots. These roots re not ll necessrily distinct. If one root r, of polynomil occurs just once, then it is clled simple root. If root r occurs exctly m times, where m >, then it is clled root of multiplicity m (or n m-fold root). If m =, r is clled double root, if m =, it is clled triple root; nd so on. If polynomil with rtionl coefficients hs the irrtionl root + numbers) then its irrtionl conjugte, b, is lso root. b (where nd b re rtionl If polynomil with rel coefficients hs the imginry root + numbers) then its imginry conjugte, b, is lso root. b (where nd b re rel A polynomil hs 0 s root if nd only if the constnt term of the eqution is zero. If the leding coefficient of polynomil is, then ll of its rtionl roots re integers. If rtionl frction q p, expressed in lowest terms, is root of polynomil (or polynomil eqution) then p is divisor of the constnt term 0 nd q is divisor of the leding coefficient, n. Notice tht the converse of this sttement is not true. For exmple, we cn only sy tht is possible root of the polynomil 9x 4 5x + 8x + 4 (since is fctor of 4 nd is fctor of 9). Unfortuntely, this cn necessitte lengthy process of exmining mny potentil roots of polynomil before one or more is ctully identified. Once we hve found one root of polynomil, it usully becomes esier to identify its remining roots. As well, there re number of useful hints nd strtegies tht cn speed up this process (but you will hve to ttend the session on Sturdy to lern these). One such rule (tht will not be delt with on Sturdy) is briefly described here. It is clled Descrtes Rule of Signs for Polynomil Roots. It first requires the definition of the vrition of sign of polynomil. When polynomil is rrnged in descending order of powers of the vrible, if two successive terms differ in sign, the polynomil is sid to hve vrition of sign. For exmple, the polynomil 4x 5 x + 5x + x 4 hs three vritions of sign. Descrtes Rule sttes tht the number of positive roots of polynomil is equl either to the number of vritions of sign or to tht number diminished by n even number. (It is not ll tht helpful for fctoring polynomils t the high school level is it!) Ech ttempt mde to confirm tht possible root (x ) for polynomil p(x) relly is root (using either long division or synthetic division) is clled tril. If it turns out tht (x ) is root of p(x) then it is clled successful tril, nd it is clled flse tril if it is unsuccessful. Once you hve identified one root of the polynomil eqution p(x) = 0, nd hve used long or synthetic division to find q(x), the remining fctor of p(x) (i.e. p(x) = (x )q(x) = 0), then the resulting eqution, q(x) = 0 is clled the depressed eqution for p(x) = 0. The first few expnsions of the binomil theorem (x + y) n re often required in working with polynomils, nd lso led to the definitions of certin types of fctoring questions. These re: (x + y) = x + xy + y nd (x + y) = x + x y + xy + y nd (x + y) 4 = x 4 + 4x y + 6x y + 4xy + y 4 These led to the definitions or formuls for fctoring perfect squre trinomils nd cn lso extend to fctoring the difference of squres nd the sum nd difference of cubes. We hve: x y = (x y)(x + y) nd x y = (x y)(x + xy + y ) nd x + y = (x y)(x xy + y )

3 Some of the different methods used to fctor polynomils likely include: Grde 9 0 T Common fctor Common fctor Common fctor Common fctor e Difference of squres Difference of squres Difference of squres c Perfect squre trinomils Perfect squre trinomils Perfect squre trinomils h Simple trinomils Grouping Sum of cubes n Complex trinomils Simple trinomils Difference of cubes i Complex trinomils Grouping q Simple trinomils u Complex trinomils e The Fctor Theorem The bsic skills in mthemtics needed to pply the techniques include strong working knowledge of: The multipliction nd division fcts up to (or to 5 5 or even to 0 0???) Perfect squres up to (or up to 5 or even up to 0 ) Perfect cubes up to 0 Here is drill sheet tht cn be used for prcticing the rpid fctoring of monic qudrtic trinomils:.) (x )(x ).) (x )(x ).) (x )(x ) 4.) (x )(x ) 5.) (x )(x ) 6.) (x )(x ) 7.) (x )(x ) 8.) (x )(x ) 9.) (x )(x ) 0.) (x )(x ) You cn work out the following questions hed of the session if you like but it is certinly not necessry to do so. Note tht there is not enough spce on this pge to fully fctor the longer types..) Fctor the following trinomils: ) x + x + 4 Sign Pttern (designted s simple bove) Question Answer b) x x c) x + x 4 + } one + nd } one sign d) x x 4 (lrgest number hs sign of the x-term).) Fctor the following trinomils: ) 5x + 6x + (designted s complex bove) b) 4x 9x + c) 6x + x 5 d) 6x 40x + 5 e) 6x 58x + 5

4 .) Fctor the following trinomils: ) 5x y 5 + 5x 4 y + 5x y (common fctoring required) b) 4x(7m 5) 8y(5 7m) c) πr + 4πrh x x x x d) e + ( x ) e e) 5x 4y 7 9z + 4y y 5 f) ( ) ( ) 6 x 4 ( 4) ( ) 6 x + x + x x + x 4.) Fctor the following trinomils: ) 5x 69y (difference of squres or cubes) b) Ω (v ) c) 4x 0x + 5 9x + 4x 49 d) x 6 y 6 e) 5x + 4 (Use difference of squres) Evlute: ) Divide: x 4 5x + 6x 4 by x using ) forml long division or b) synthetic division (Horner s Method for evluting function in x, given x = ) ) Fctor the following trinomils: ) x x 0x + 4 (fctor theorem my be required) b) x + 9x + x + 5 c) x + 0x 8x 40 d) 6x 40x 5 7.) Solve for x: ) x = x + b) 8x 4 8x x + = 0 (Do you recognize the significnce of these two equtions?)

5 Note tht φ = + 5 (See the ttched Excel sheet on the Golden Men.) The Qudrtic Formul: x = b ± b 4c The generl formul for solving the cubic eqution x + bx + cx + d = 0 is given by: 4 { b + [ b + 9bc 7d + b c + 8 d 54 bcd + b d + ] x = c 4 [ b + 9bc 7d b c + 8 d 54 bcd + b d + ] } + c A Closer Look t The Difference of Squres The following re in incresing order of difficulty or complexity (s you go down, then cross) x 6 x 4 5 (x 5) (y + ) 4x + x + 9 y 6y 9 9x 6 x 6 5y 0 9(x 5) 6(y + ) 6x + 40x + 5 9x 48y 64 9x + 6 x 4 8 x + 0x y 4x + 6z + 5y z 9 + 0xy 9x 6y (x + 5) 6 9y x + 6x 9 x 4 + 4x + 6 4x 54 6 (x 5)

6 Some Selected Number Squring Ptterns Squring Numbers Ending In 5 Squring Numbers in the 50 s Squring Numbers in the 00 s Squring Numbers in the 90 s 5 = 5 50 = = = = 65 5 = 60 0 = = = 5 5 = = = = 5 = = 96 = = 54 = 05 = 95 = 65 = 55 = 06 = 94 = 75 = 56 = 07 = 9 = 85 = 57 = 08 = 9 = 95 = 58 = 09 = 9 = 995 = 59 = 4 = 90 = Why the First Two Ptterns Work.) e.g. 57 = Let the generl term be 50 + n, n ε I + Now (50 + n) = 50(50 + n) + n(50 + n) = n + 50n + n = n + n = 00(5 + n) + n.) e.g. 65 = 00 (6 7) + 5 Let the generl number be 0n + 5, n ε I + Now (0n + 5) = 0n(0n + 5) + 5(0n + 5) = 00n + 50n + 50n + 5 = 00n + 00n +5 = 00(n + n) + 5 = 00(n)(n+) + 5

7 Divisibility Tests Dividing by Look t the ones digit. If it is divisible by, then the number is s well. Exmple: is divisible by four lso, becuse 6 is divisible by two. Dividing by Add up the digits: if the sum is divisible by three, then the number is s well. Exmples: : the digits dd to 6 so the whole number is divisible by three The digits dd up to 57, nd 5 +7 =, so the originl number is divisible by three. Dividing by 4 Look t the lst two digits (the tens nd ones digits). If they re divisible by 4, the number is s well. Exmple: is divisible by four lso, becuse is divisible by four. Dividing by 5 If the ones digit is five or zero, then the number is divisible by 5. Dividing by 6 Check the rule for divisibility by nd by. If the number is divisible by both nd, it is divisible by 6 s well. Dividing by 7 To find out if number is divisible by seven, tke the ones digit, double it, nd subtrct it from the rest of the number. Exmple: If you hd 0, you would double the ones digit to get six, nd subtrct tht from 0 to get 4. If you get n nswer divisible by 7 (including zero), then the originl number is divisible by seven. If you don't know the new number's divisibility, you cn pply the rule gin. Here is nother method for divisibility by seven: Divide off the number into groups of digits (from the right). You my be left with group of one or two digits on the left end. Then find the sum of the lternte groups of these new -digit numbers. If the difference between the two sums is divisible by 7, then so ws the originl number.

8 Exmple: For the number divide up the number into these groups of digits Now dd up the lternte groups: = 5 nd = 67 Find the difference between theses new sums: 5 67 = 64 Since 64 is evenly divisible by 7 (5 times) then lso ws divisible by 7. Dividing by 8 Check the lst three digits (the hundreds, tens nd ones digits). If the lst three digits of number re divisible by 8, then so is the whole number. Exmple: 888 is divisible by 8; 886 isn't. Dividing by 9 Add the digits. If they re divisible by nine, then the number is s well. This holds for ny power of three. Dividing by 0 If the units digit is 0, then the number is divisible by 0. Dividing by Tke ny number, such s Add the first, third, fifth, seventh,.., digits = Add the second, fourth, sixth, eighth,.., digits = If the difference, including 0, is divisible by, then so is the number. - = 0 so is evenly divisible by. Dividing by Check for divisibility by nd 4. Dividing by Delete the ones digit from the given number. Then subtrct nine times the deleted digit from the remining number. If wht is left is divisible by, then so is the originl number. Exmple: For the number 4667, delete lst 7 nd multiply it by 9 (7 9 = 6). Now = 40. Since 40 is divisible by ( times) so too is Another method for divisibility tests for nd for involves exctly the sme process s the second method given for divisibility by 7. Divide the number into groups of three nd find the sum of lternting groups. If the difference of these sums is divisible by (or by ) then the originl number ws lso divisible by (or ).

9 Let x = t u (where t > u) Consider squre ABCD with side of length t (shown t right) Let u be the length of the side of smller squre (shded in blue in the digrm t right. Thus the pink squre (shown t right) hs sides of length t u. We lso hve two rectngles (shown in An Alternte Proof for The Qudrtic Formul t u A t u u B t u u t u t u u yellow), ech with length t u D t u u C nd width u. t The re of lrge squre (with side of length t) equls the sum of the res of the two smller (in pink nd blue) squres nd the two (yellow) rectngles. Thus we hve: t = (t u) + u(t u) + u which gives: t u = (t u) + u(t u) Now let m = u nd n = t u Hence, u = m nd thus t u = Therefore x = t u = m ± + 4 t m = n Therefore t m + n 4 n m 4 m = ± m + n = Since t u = (t u) + u(t u) we get (t u) = t u u(t u) = which gives m t = ± n m ± m + 4n or x = n mx which gives us x + mx n = 0 nd the solution for x in this eqution is: x = m ± m + 4n Of course, x + mx n = 0 cn be expressed s x + bx + c = 0 if we let This would mke the solution for x + bx + c = 0 result in x = b ± b 4c = b ± b 4c b ± = b 4c m = b nd n = c b ± b 4c x = which gives

10 Hints to use to identify possible roots when using the Fctor Theorem (In the exmples below, we men tht if (x ) is fctor then is root).) For the polynomil n x n + n x n + n x n + x + 0 (or the corresponding polynomil eqution): ) If p is divisor of the constnt term 0 then ± p is possible divisor of the polynomil, nd b) If q is divisor of the leding coefficient, n then qx ± p is possible fctor of the polynomil For exmple, the polynomil x + 6x 8x 0 could hve roots ±, ±, ±5, ±0 ±, ±, ± 5 nd ± 0. (You must still test one or more of these roots until you obtin successful tril.).) If the signs lternte from term to term in the polynomil (including the signs of terms with coefficient zero), then t lest one root is positive. Exmples of this type of polynomil include: 5x 9x + 7x 5 nd x 4 + 6x 8x + 5 (note tht it must be written: x 4 0x + 6x 8x + 5).) If the signs of every term of the polynomil re positive, then ll of the rtionl roots of the polynomil re negtive. e.g. ll of the rtionl roots of x 4 + 5x + 7x + 8x + 4 re negtive 4.) If the bsolute vlue of the lrgest coefficient of ny term of the polynomil is lrger thn the sum of the bsolute vlues of ech of the remining terms, then neither nor is root of tht polynomil. e.g. in the polynomil x +6x 5x 4, neither nor cn be root (becuse 6 > ) 5.) If the sum of the coefficients of the terms of polynomil re zero, then is root of tht polynomil. e.g. is root of x 4 9x + 4x + 8x 4 becuse = 0 6.) This one is little more difficult to see t first glnce, nd it relly only works becuse the text book uthors re mking up questions tht cn be fctored. If you re ble to mke the sum of the coefficients of ll terms of the polynomil zero by chnging one or more signs, then is root of the polynomil. Exmples of this type re: x 4 + 5x 7x + 9x + 4 (becuse equls 0) nd x 4 x + 8x + x + 5 (becuse = 0). 7.) Descrtes Rule of Signs for Polynomil Roots. This first requires the definition of the vrition of sign of polynomil. When polynomil is rrnged in descending order of powers of the vrible, if two successive terms differ in sign, the polynomil is sid to hve vrition of sign. For exmple, the polynomil 4x 5 x + 5x + x 4 hs three vritions of sign. Descrtes Rule sttes tht the number of positive roots of polynomil is equl either to the number of vritions of sign or to tht number diminished by n even number. Be wre tht it is not necessry to insert missing terms with zero coefficient (such s 0x 4 in the exmple bove) when using this rule, but it is necessry when using the rule in point (.) given bove. The suggested method for testing potentil roots in the quickest possible fshion is to use synthetic division. This method not only distinguishes between flse nd successful trils fster thn cn be obtined by using clcultor, but it lso identifies the depressed fctor (i.e. the other fctor).

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

Solving BAMO Problems

Solving BAMO Problems Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

Answer, Key Homework 10 David McIntyre 1

Answer, Key Homework 10 David McIntyre 1 Answer, Key Homework 10 Dvid McIntyre 1 This print-out should hve 22 questions, check tht it is complete. Multiple-choice questions my continue on the next column or pge: find ll choices efore mking your

More information

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives

More information

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2 7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

More information

Helicopter Theme and Variations

Helicopter Theme and Variations Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

More information

PHY 140A: Solid State Physics. Solution to Homework #2

PHY 140A: Solid State Physics. Solution to Homework #2 PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

Factoring RSA moduli with weak prime factors

Factoring RSA moduli with weak prime factors Fctoring RSA moduli with we prime fctors Abderrhmne Nitj 1 nd Tjjeeddine Rchidi 2 1 Lbortoire de Mthémtiques Nicols Oresme Université de Cen Bsse Normndie, Frnce bderrhmne.nitj@unicen.fr 2 School of Science

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 4721 Money and Banking Problem Set 2 Answer Key Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

More information

#A12 INTEGERS 13 (2013) THE DISTRIBUTION OF SOLUTIONS TO XY = N (MOD A) WITH AN APPLICATION TO FACTORING INTEGERS

#A12 INTEGERS 13 (2013) THE DISTRIBUTION OF SOLUTIONS TO XY = N (MOD A) WITH AN APPLICATION TO FACTORING INTEGERS #A1 INTEGERS 13 (013) THE DISTRIBUTION OF SOLUTIONS TO XY = N (MOD A) WITH AN APPLICATION TO FACTORING INTEGERS Michel O. Rubinstein 1 Pure Mthemtics, University of Wterloo, Wterloo, Ontrio, Cnd mrubinst@uwterloo.c

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

Introduction to Integration Part 2: The Definite Integral

Introduction to Integration Part 2: The Definite Integral Mthemtics Lerning Centre Introduction to Integrtion Prt : The Definite Integrl Mr Brnes c 999 Universit of Sdne Contents Introduction. Objectives...... Finding Ares 3 Ares Under Curves 4 3. Wht is the

More information

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time

More information

Second-Degree Equations as Object of Learning

Second-Degree Equations as Object of Learning Pper presented t the EARLI SIG 9 Biennil Workshop on Phenomenogrphy nd Vrition Theory, Kristinstd, Sweden, My 22 24, 2008. Abstrct Second-Degree Equtions s Object of Lerning Constnt Oltenu, Ingemr Holgersson,

More information

MULTIPLYING OUT & FACTORING

MULTIPLYING OUT & FACTORING igitl ircuit Engineering MULTIPLYING OUT & FTORING I IGITL SIGN Except for #$&@ fctoring st istributive X + X = X( + ) 2nd istributive (X + )(X + ) = X + (X + )(X + )(X + ) = X + Swp (X + )(X + ) = X +

More information

Euler Euler Everywhere Using the Euler-Lagrange Equation to Solve Calculus of Variation Problems

Euler Euler Everywhere Using the Euler-Lagrange Equation to Solve Calculus of Variation Problems Euler Euler Everywhere Using the Euler-Lgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

All pay auctions with certain and uncertain prizes a comment

All pay auctions with certain and uncertain prizes a comment CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

Pure C4. Revision Notes

Pure C4. Revision Notes Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd

More information

Applications to Physics and Engineering

Applications to Physics and Engineering Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics

More information

Small Businesses Decisions to Offer Health Insurance to Employees

Small Businesses Decisions to Offer Health Insurance to Employees Smll Businesses Decisions to Offer Helth Insurnce to Employees Ctherine McLughlin nd Adm Swinurn, June 2014 Employer-sponsored helth insurnce (ESI) is the dominnt source of coverge for nonelderly dults

More information

MATH PLACEMENT REVIEW GUIDE

MATH PLACEMENT REVIEW GUIDE MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

Redistributing the Gains from Trade through Non-linear. Lump-sum Transfers

Redistributing the Gains from Trade through Non-linear. Lump-sum Transfers Redistributing the Gins from Trde through Non-liner Lump-sum Trnsfers Ysukzu Ichino Fculty of Economics, Konn University April 21, 214 Abstrct I exmine lump-sum trnsfer rules to redistribute the gins from

More information

Quality Evaluation of Entrepreneur Education on Graduate Students Based on AHP-fuzzy Comprehensive Evaluation Approach ZhongXiaojun 1, WangYunfeng 2

Quality Evaluation of Entrepreneur Education on Graduate Students Based on AHP-fuzzy Comprehensive Evaluation Approach ZhongXiaojun 1, WangYunfeng 2 Interntionl Journl of Engineering Reserch & Science (IJOER) ISSN [2395-6992] [Vol-2, Issue-1, Jnury- 2016] Qulity Evlution of Entrepreneur Eduction on Grdute Students Bsed on AHP-fuzzy Comprehensive Evlution

More information

CURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line.

CURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line. CURVES ANDRÉ NEVES 1. Problems (1) (Ex 1 of 1.3 of Do Crmo) Show tht the tngent line to the curve α(t) (3t, 3t 2, 2t 3 ) mkes constnt ngle with the line z x, y. (2) (Ex 6 of 1.3 of Do Crmo) Let α(t) (e

More information

On the Robustness of Most Probable Explanations

On the Robustness of Most Probable Explanations On the Robustness of Most Probble Explntions Hei Chn School of Electricl Engineering nd Computer Science Oregon Stte University Corvllis, OR 97330 chnhe@eecs.oregonstte.edu Adnn Drwiche Computer Science

More information

SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1470 - COLLEGE ALGEBRA (4 SEMESTER HOURS)

SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1470 - COLLEGE ALGEBRA (4 SEMESTER HOURS) SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 470 - COLLEGE ALGEBRA (4 SEMESTER HOURS). COURSE DESCRIPTION: Polynomil, rdicl, rtionl, exponentil, nd logrithmic functions

More information

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, cross-clssified

More information

C-crcs Cognitive - Counselling Research & Conference Services (eissn: 2301-2358)

C-crcs Cognitive - Counselling Research & Conference Services (eissn: 2301-2358) C-crcs Cognitive - Counselling Reserch & Conference Services (eissn: 2301-2358) Volume I Effects of Music Composition Intervention on Elementry School Children b M. Hogenes, B. Vn Oers, R. F. W. Diekstr,

More information

STATUS OF LAND-BASED WIND ENERGY DEVELOPMENT IN GERMANY

STATUS OF LAND-BASED WIND ENERGY DEVELOPMENT IN GERMANY Yer STATUS OF LAND-BASED WIND ENERGY Deutsche WindGurd GmbH - Oldenburger Strße 65-26316 Vrel - Germny +49 (4451)/9515 - info@windgurd.de - www.windgurd.com Annul Added Cpcity [MW] Cumultive Cpcity [MW]

More information

Week 11 - Inductance

Week 11 - Inductance Week - Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n

More information

Numerical Methods of Approximating Definite Integrals

Numerical Methods of Approximating Definite Integrals 6 C H A P T E R Numericl Methods o Approimting Deinite Integrls 6. APPROXIMATING SUMS: L n, R n, T n, AND M n Introduction Not only cn we dierentite ll the bsic unctions we ve encountered, polynomils,

More information

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a. Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

More information

Words Symbols Diagram. abcde. a + b + c + d + e

Words Symbols Diagram. abcde. a + b + c + d + e Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

More information

baby on the way, quit today

baby on the way, quit today for mums-to-be bby on the wy, quit tody WHAT YOU NEED TO KNOW bout smoking nd pregnncy uitting smoking is the best thing you cn do for your bby We know tht it cn be difficult to quit smoking. But we lso

More information

body.allow-sidebar OR.no-sidebar.home-page (if this is the home page).has-custom-banner OR.nocustom-banner .IR OR.no-IR

body.allow-sidebar OR.no-sidebar.home-page (if this is the home page).has-custom-banner OR.nocustom-banner .IR OR.no-IR body.llow-sidebr OR.no-sidebr.home-pge (if this is the home pge).hs-custom-bnner OR.nocustom-bnner.IR OR.no-IR #IDENTIFIER_FOR_THIS_SITE div#pge-continer.depends_on_page_ty PE llow-sidebr mens tht there

More information

Online Multicommodity Routing with Time Windows

Online Multicommodity Routing with Time Windows Konrd-Zuse-Zentrum für Informtionstechnik Berlin Tkustrße 7 D-14195 Berlin-Dhlem Germny TOBIAS HARKS 1 STEFAN HEINZ MARC E. PFETSCH TJARK VREDEVELD 2 Online Multicommodity Routing with Time Windows 1 Institute

More information

Small Business Cloud Services

Small Business Cloud Services Smll Business Cloud Services Summry. We re thick in the midst of historic se-chnge in computing. Like the emergence of personl computers, grphicl user interfces, nd mobile devices, the cloud is lredy profoundly

More information

Small Business Networking

Small Business Networking Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer

More information

Network Configuration Independence Mechanism

Network Configuration Independence Mechanism 3GPP TSG SA WG3 Security S3#19 S3-010323 3-6 July, 2001 Newbury, UK Source: Title: Document for: AT&T Wireless Network Configurtion Independence Mechnism Approvl 1 Introduction During the lst S3 meeting

More information

Labor Productivity and Comparative Advantage: The Ricardian Model of International Trade

Labor Productivity and Comparative Advantage: The Ricardian Model of International Trade Lbor Productivity nd omrtive Advntge: The Ricrdin Model of Interntionl Trde Model of trde with simle (unrelistic) ssumtions. Among them: erfect cometition; one reresenttive consumer; no trnsction costs,

More information

SOLVING EQUATIONS BY FACTORING

SOLVING EQUATIONS BY FACTORING 316 (5-60) Chpter 5 Exponents nd Polynomils 5.9 SOLVING EQUATIONS BY FACTORING In this setion The Zero Ftor Property Applitions helpful hint Note tht the zero ftor property is our seond exmple of getting

More information

Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00

Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00 COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided

More information

persons withdrawing from addiction is given by summarizing over individuals with different ages and numbers of years of addiction remaining:

persons withdrawing from addiction is given by summarizing over individuals with different ages and numbers of years of addiction remaining: COST- BENEFIT ANALYSIS OF NARCOTIC ADDICTION TREATMENT PROGRAMS with Specil Reference to Age Irving Leveson,l New York City Plnning Commission Introduction Efforts to del with consequences of poverty,

More information

AAPT UNITED STATES PHYSICS TEAM AIP 2010

AAPT UNITED STATES PHYSICS TEAM AIP 2010 2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS - 75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD

More information

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999 Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

The Impact of Oligopolistic Competition in Networks

The Impact of Oligopolistic Competition in Networks OPERATIONS RESEARCH Vol. 57, No. 6, November December 2009, pp. 1421 1437 issn 0030-364X eissn 1526-5463 09 5706 1421 informs doi 10.1287/opre.1080.0653 2009 INFORMS The Impct of Oligopolistic Competition

More information

Space Vector Pulse Width Modulation Based Induction Motor with V/F Control

Space Vector Pulse Width Modulation Based Induction Motor with V/F Control Interntionl Journl of Science nd Reserch (IJSR) Spce Vector Pulse Width Modultion Bsed Induction Motor with V/F Control Vikrmrjn Jmbulingm Electricl nd Electronics Engineering, VIT University, Indi Abstrct:

More information

Contextualizing NSSE Effect Sizes: Empirical Analysis and Interpretation of Benchmark Comparisons

Contextualizing NSSE Effect Sizes: Empirical Analysis and Interpretation of Benchmark Comparisons Contextulizing NSSE Effect Sizes: Empiricl Anlysis nd Interprettion of Benchmrk Comprisons NSSE stff re frequently sked to help interpret effect sizes. Is.3 smll effect size? Is.5 relly lrge effect size?

More information

icbs: Incremental Cost based Scheduling under Piecewise Linear SLAs

icbs: Incremental Cost based Scheduling under Piecewise Linear SLAs i: Incrementl Cost bsed Scheduling under Piecewise Liner SLAs Yun Chi NEC Lbortories Americ 18 N. Wolfe Rd., SW3 35 Cupertino, CA 9514, USA ychi@sv.nec lbs.com Hyun Jin Moon NEC Lbortories Americ 18 N.

More information

Version 001 Summer Review #03 tubman (IBII20142015) 1

Version 001 Summer Review #03 tubman (IBII20142015) 1 Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This print-out should he 35 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03

More information

Lower Bound for Envy-Free and Truthful Makespan Approximation on Related Machines

Lower Bound for Envy-Free and Truthful Makespan Approximation on Related Machines Lower Bound for Envy-Free nd Truthful Mespn Approximtion on Relted Mchines Lis Fleischer Zhenghui Wng July 14, 211 Abstrct We study problems of scheduling jobs on relted mchines so s to minimize the mespn

More information

Decision Rule Extraction from Trained Neural Networks Using Rough Sets

Decision Rule Extraction from Trained Neural Networks Using Rough Sets Decision Rule Extrction from Trined Neurl Networks Using Rough Sets Alin Lzr nd Ishwr K. Sethi Vision nd Neurl Networks Lbortory Deprtment of Computer Science Wyne Stte University Detroit, MI 48 ABSTRACT

More information

INTERCHANGING TWO LIMITS. Zoran Kadelburg and Milosav M. Marjanović

INTERCHANGING TWO LIMITS. Zoran Kadelburg and Milosav M. Marjanović THE TEACHING OF MATHEMATICS 2005, Vol. VIII, 1, pp. 15 29 INTERCHANGING TWO LIMITS Zorn Kdelburg nd Milosv M. Mrjnović This pper is dedicted to the memory of our illustrious professor of nlysis Slobodn

More information

Data replication in mobile computing

Data replication in mobile computing Technicl Report, My 2010 Dt repliction in mobile computing Bchelor s Thesis in Electricl Engineering Rodrigo Christovm Pmplon HALMSTAD UNIVERSITY, IDE SCHOOL OF INFORMATION SCIENCE, COMPUTER AND ELECTRICAL

More information

6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions 6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

More information

Techniques for Requirements Gathering and Definition. Kristian Persson Principal Product Specialist

Techniques for Requirements Gathering and Definition. Kristian Persson Principal Product Specialist Techniques for Requirements Gthering nd Definition Kristin Persson Principl Product Specilist Requirements Lifecycle Mngement Elicit nd define business/user requirements Vlidte requirements Anlyze requirements

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

AP STATISTICS SUMMER MATH PACKET

AP STATISTICS SUMMER MATH PACKET AP STATISTICS SUMMER MATH PACKET This pcket is review of Algebr I, Algebr II, nd bsic probbility/counting. The problems re designed to help you review topics tht re importnt to your success in the clss.

More information

Anthem Blue Cross Life and Health Insurance Company University of Southern California Custom Premier PPO 800/20%/20%

Anthem Blue Cross Life and Health Insurance Company University of Southern California Custom Premier PPO 800/20%/20% Anthem Blue Cross Life nd Helth Insurnce Compny University of Southern Cliforni Custom Premier 800/20%/20% Summry of Benefits nd Coverge: Wht this Pln Covers & Wht it Costs Coverge Period: 01/01/2015-12/31/2015

More information

4. DC MOTORS. Understand the basic principles of operation of a DC motor. Understand the operation and basic characteristics of simple DC motors.

4. DC MOTORS. Understand the basic principles of operation of a DC motor. Understand the operation and basic characteristics of simple DC motors. 4. DC MOTORS Almost every mechnicl movement tht we see round us is ccomplished by n electric motor. Electric mchines re mens o converting energy. Motors tke electricl energy nd produce mechnicl energy.

More information

SOLVING QUADRATIC EQUATIONS BY FACTORING

SOLVING QUADRATIC EQUATIONS BY FACTORING 6.6 Solving Qudrti Equtions y Ftoring (6 31) 307 In this setion The Zero Ftor Property Applitions 6.6 SOLVING QUADRATIC EQUATIONS BY FACTORING The tehniques of ftoring n e used to solve equtions involving

More information

AntiSpyware Enterprise Module 8.5

AntiSpyware Enterprise Module 8.5 AntiSpywre Enterprise Module 8.5 Product Guide Aout the AntiSpywre Enterprise Module The McAfee AntiSpywre Enterprise Module 8.5 is n dd-on to the VirusScn Enterprise 8.5i product tht extends its ility

More information

I calculate the unemployment rate as (In Labor Force Employed)/In Labor Force

I calculate the unemployment rate as (In Labor Force Employed)/In Labor Force Introduction to the Prctice of Sttistics Fifth Edition Moore, McCbe Section 4.5 Homework Answers to 98, 99, 100,102, 103,105, 107, 109,110, 111, 112, 113 Working. In the lnguge of government sttistics,

More information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4) ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

More information

Harvard College. Math 21a: Multivariable Calculus Formula and Theorem Review

Harvard College. Math 21a: Multivariable Calculus Formula and Theorem Review Hrvrd College Mth 21: Multivrible Clculus Formul nd Theorem Review Tommy McWillim, 13 tmcwillim@college.hrvrd.edu December 15, 2009 1 Contents Tble of Contents 4 9 Vectors nd the Geometry of Spce 5 9.1

More information

Health insurance marketplace What to expect in 2014

Health insurance marketplace What to expect in 2014 Helth insurnce mrketplce Wht to expect in 2014 33096VAEENBVA 06/13 The bsics of the mrketplce As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum

More information

Recognition Scheme Forensic Science Content Within Educational Programmes

Recognition Scheme Forensic Science Content Within Educational Programmes Recognition Scheme Forensic Science Content Within Eductionl Progrmmes one Introduction The Chrtered Society of Forensic Sciences (CSoFS) hs been ccrediting the forensic content of full degree courses

More information

Basic Research in Computer Science BRICS RS-02-13 Brodal et al.: Solving the String Statistics Problem in Time O(n log n)

Basic Research in Computer Science BRICS RS-02-13 Brodal et al.: Solving the String Statistics Problem in Time O(n log n) BRICS Bsic Reserch in Computer Science BRICS RS-02-13 Brodl et l.: Solving the String Sttistics Prolem in Time O(n log n) Solving the String Sttistics Prolem in Time O(n log n) Gerth Stølting Brodl Rune

More information

Is Competition Among Charities Bad?

Is Competition Among Charities Bad? Is Competition Among Chrities Bd? Inkyung Ch nd Willim Neilson Tes A&M University, College Sttion, TX 7783 December Abstrct This pper studies tht the eect o incresed competition mong chrities or dontions,

More information

Ostrowski Type Inequalities and Applications in Numerical Integration. Edited By: Sever S. Dragomir. and. Themistocles M. Rassias

Ostrowski Type Inequalities and Applications in Numerical Integration. Edited By: Sever S. Dragomir. and. Themistocles M. Rassias Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology,

More information

UNITED STATES DEPARTMENT OF AGRICULTURE Washington, D.C. 20250. ACTION BY: All Divisions and Offices. FGIS Directive 2510.

UNITED STATES DEPARTMENT OF AGRICULTURE Washington, D.C. 20250. ACTION BY: All Divisions and Offices. FGIS Directive 2510. UNITED STATES DEPARTMENT OF AGRICULTURE Wshington, D.C. 20250 ACTION BY: All Divisions nd Offices FGIS Directive 2510.1 12-11-73 FEDERAL TORT CLAIMS I PURPOSE This Instruction: A Sets forth the bsic provisions

More information

Hillsborough Township Public Schools Mathematics Department Computer Programming 1

Hillsborough Township Public Schools Mathematics Department Computer Programming 1 Essentil Unit 1 Introduction to Progrmming Pcing: 15 dys Common Unit Test Wht re the ethicl implictions for ming in tody s world? There re ethicl responsibilities to consider when writing computer s. Citizenship,

More information

Health insurance exchanges What to expect in 2014

Health insurance exchanges What to expect in 2014 Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 02/13 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount

More information

Value Function Approximation using Multiple Aggregation for Multiattribute Resource Management

Value Function Approximation using Multiple Aggregation for Multiattribute Resource Management Journl of Mchine Lerning Reserch 9 (2008) 2079-2 Submitted 8/08; Published 0/08 Vlue Function Approximtion using Multiple Aggregtion for Multittribute Resource Mngement Abrhm George Wrren B. Powell Deprtment

More information