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


 Aileen Hicks
 1 years ago
 Views:
Transcription
1 33337_0P03.qp 2/27/ :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 lgeric epression is the polynomil. Some emples re 2 5, , nd 5 2y 2 y 3. The first two re polynomils in nd the third is polynomil in nd y. The terms of polynomil in hve the form k, where is the coefficient nd k is the degree of the term. For instnce, the polynomil 共 5兲 2 共0兲 Let 0,, 2,..., n e rel numers nd let n e nonnegtive integer. A polynomil in is n epression of the form P.3 n n Write polynomils in stndrd form. Add, sutrct, nd multiply polynomils. Use specil products to multiply polynomils. Remove common fctors from polynomils. Fctor specil polynomil forms. Fctor trinomils s the product of two inomils. Fctor y grouping. Why you should lern it Definition of Polynomil in n hs coefficients 2, 5, 0, nd. n Polynomils cn e used to model nd solve rellife prolems. For instnce, in Eercise 57 on pge 34, polynomil is used to model the totl distnce n utomoile trvels when stopping where n 0. The polynomil is of degree n, n is the leding coefficient, nd 0 is the constnt term. In stndrd form, polynomil in is written with descending powers of. Polynomils with one, two, nd three terms re clled monomils, inomils, nd trinomils, respectively. A polynomil tht hs ll zero coefficients is clled the zero polynomil, denoted y 0. No degree is ssigned to this prticulr polynomil. For polynomils in more thn one vrile, the degree of term is the sum of the eponents of the vriles in the term. The degree of the polynomil is the highest degree of its terms. For instnce, the degree of the polynomil 23y6 4y 7y4 is ecuse the sum of the eponents in the lst term is the gretest. Epressions such s the following re not polynomils. 3 冪3 3 共3兲兾 Roert W. Ginn/ge fotostock The eponent 兾2 is not n integer. The eponent is not nonnegtive integer. Emple Writing Polynomils in Stndrd Form Polynomil Stndrd Form c. 8 8 共8 8 0兲 Now try Eercise 5. Degree STUDY TIP Epressions re not polynomils if:. A vrile is underneth rdicl. 2. A polynomil epression (with degree greter thn 0) is in the denomintor of term.
2 33337_0P03.qp 2/27/06 9:3 AM Pge 25 Opertions with Polynomils You cn dd nd sutrct polynomils in much the sme wy you dd nd sutrct rel numers. Simply dd or sutrct the like terms (terms hving ectly the sme vriles to ectly the sme powers) y dding their coefficients. For instnce, 3y 2 nd 5y 2 re like terms nd their sum is 3y 2 5y y 2 2y 2. Section P.3 Polynomils nd Fctoring 25 Emple 2 Sums nd Differences of Polynomils Perform the indicted opertion Now try Eercise 23. Group like terms. Comine like terms. Distriutive Property Group like terms. Comine like terms. STUDY TIP When negtive sign precedes n epression within prentheses, tret it like the coefficient nd distriute the negtive sign to ech term inside the prentheses To find the product of two polynomils, use the left nd right Distriutive Properties. Emple 3 Multiplying Polynomils: The FOIL Method When using the FOIL method, the following scheme my e helpful. F L Product of First terms Product of Outer terms Product of Inner terms Product of Lst terms I O Note tht when using the FOIL Method (which cn e used only to multiply two inomils), the outer (O) nd inner (I) terms my e like terms tht cn e comined into one term. Now try Eercise 39.
3 33337_0P03.qp 2/27/06 9:3 AM Pge Chpter P Prerequisites Emple 4 The Product of Two Trinomils Find the product of nd When multiplying two polynomils, e sure to multiply ech term of one polynomil y ech term of the other. A verticl formt is helpful Now try Eercise 59. Write in stndrd form. Write in stndrd form Comine like terms. Specil Products Specil Products Let u nd v e rel numers, vriles, or lgeric epressions. Specil Product Sum nd Difference of Sme Terms u v u v u 2 v 2 Squre of Binomil Cue of Binomil Emple u v 2 u 2 2uv v u v 2 u 2 2uv v u v 3 u 3 3u 2 v 3uv 2 v u v 3 u 3 3u 2 v 3uv 2 v Emple 5 The Product of Two Trinomils Find the product of y 2 nd y 2. By grouping y in prentheses, you cn write the product of the trinomils s specil product. y 2 y 2 y 2 y 2 y y y 2 4 Now try Eercise 6. To understnd the individul ptterns of specil products, hve students derive ech product. Then eplin how specil products sve time nd how the pttern of the product must e recognized to fctor n epression.
4 33337_0P03.qp 2/27/06 9:3 AM Pge 27 Section P.3 Polynomils nd Fctoring 27 Fctoring The process of writing polynomil s product is clled fctoring. It is n importnt tool for solving equtions nd for simplifying rtionl epressions. Unless noted otherwise, when you re sked to fctor polynomil, you cn ssume tht you re looking for fctors with integer coefficients. If polynomil cnnot e fctored using integer coefficients, it is prime or irreducile over the integers. For instnce, the polynomil 2 3 is irreducile over the integers. Over the rel numers, this polynomil cn e fctored s A polynomil is completely fctored when ech of its fctors is prime. So, is completely fctored, ut is not completely fctored. Its complete fctoriztion is Completely fctored Not completely fctored The simplest type of fctoring involves polynomil tht cn e written s the product of monomil nd nother polynomil. The technique used here is the Distriutive Property, c c, in the reverse direction. For instnce, the polynomil cn e fctored s follows is common fctor. The first step in completely fctoring polynomil is to remove (fctor out) ny common fctors, s shown in the net emple. Activities. Eplin wht hppens when the prentheses re removed from the epression: Answer: The sign of ech term chnges Multiply using specil products: Answer: Multiply: 2 5. Answer: Emple 6 Removing Common Fctors Fctor ech epression c is common fctor is common fctor c. 2 is common fctor. Now try Eercise 73. Fctoring Specil Polynomil Forms Some polynomils hve specil forms tht rise from the specil product forms on pge 26. You should lern to recognize these forms so tht you cn fctor such polynomils esily.
5 33337_0P03.qp 2/27/06 9:3 AM Pge Chpter P Prerequisites Fctoring Specil Polynomil Forms Fctored Form Difference of Two Squres u 2 v 2 u v u v Emple Perfect Squre Trinomil u 2 2uv v 2 u v u 2 2uv v 2 u v Sum or Difference of Two Cues u 3 v 3 u v u 2 uv v 2 u 3 v 3 u v u 2 uv v One of the esiest specil polynomil forms to fctor is the difference of two squres. Think of this form s follows. u 2 v 2 u v u v Difference Opposite signs To recognize perfect squre terms, look for coefficients tht re squres of integers nd vriles rised to even powers. Emple 7 Removing Common Fctor First is common fctor Difference of two squres Fctored form Now try Eercise 77. Emple 8 Fctoring the Difference of Two Squres STUDY TIP In Emple 7, note tht the first step in fctoring polynomil is to check for common fctor. Once the common fctor is removed, it is often possile to recognize ptterns tht were not immeditely ovious y 2 2 y 2 y 2 y 2 y Difference of two squres Difference of two squres Fctored form Now try Eercise 8.
6 33337_0P03.qp 2/27/06 9:3 AM Pge 29 A perfect squre trinomil is the squre of inomil, s shown elow. u 2 2uv v 2 u v 2 or u 2 2uv v 2 u v 2 Like signs Like signs Note tht the first nd lst terms re squres nd the middle term is twice the product of u nd v. Emple 9 Fctoring Perfect Squre Trinomils Fctor ech trinomil Rewrite in u 2 2uv v 2 form Rewrite in u 2 2uv v 2 form. 4 2 Now try Eercise 87. Section P.3 Polynomils nd Fctoring 29 The net two formuls show the sums nd differences of cues. Py specil ttention to the signs of the terms. Like signs Like signs u 3 v 3 u v u 2 uv v 2 u 3 v 3 u v u 2 uv v 2 Unlike signs Unlike signs Emple 0 Fctoring the Difference of Cues Fctor Eplortion Rewrite u 6 v 6 s the difference of two squres. Then find formul for completely fctoring u 6 v 6. Use your formul to fctor completely 6 nd Now try Eercise 92. Rewrite 27 s 3 3. Fctor. Emple Fctoring the Sum of Cues Now try Eercise is common fctor. Rewrite 64 s 4 3. Fctor.
7 33337_0P03.qp 2/27/06 9:3 AM Pge Chpter P Prerequisites Trinomils with Binomil Fctors To fctor trinomil of the form 2 c, use the following pttern. Fctors of You cn drw rrows to find the correct middle term. (Encourge students to find the middle term mentlly.) c Fctors of c The gol is to find comintion of fctors of nd c such tht the outer nd inner products dd up to the middle term. For instnce, in the trinomil , you cn write ll possile fctoriztions nd determine which one hs outer nd inner products tht dd up to , 6 5, 2 3 5, You cn see tht is the correct fctoriztion ecuse the outer (O) nd inner (I) products dd up to Middle term Middle term F O I L O I Emple 2 Fctoring Trinomil: Leding Coefficient Is Fctor The possile fctoriztions re 2 6, 2, nd 3 4. Testing the middle term, you will find the correct fctoriztion to e O I Now try Eercise 03. Emple 3 Fctor Fctoring Trinomil: Leding Coefficient Is Not Point out to students tht testing the middle term mens using the FOIL method to find the sum of the outer nd inner terms. STUDY TIP Fctoring trinomil cn involve tril nd error. However, once you hve produced the fctored form, it is n esy mtter to check your nswer. For instnce, you cn verify the fctoriztion in Emple 2 y multiplying out the epression 3 4 to see tht you otin the originl trinomil, The eight possile fctoriztions re s follows. 2 5, 2 5, 2 3 5, 2 3 5, 2 5 3, 2 5 3, 2 5, 2 5 Testing the middle term, you will find the correct fctoriztion to e O I 6 5 Now try Eercise.
8 33337_0P03.qp 2/27/06 9:32 AM Pge 3 Fctoring y Grouping Sometimes polynomils with more thn three terms cn e fctored y method clled fctoring y grouping. Section P.3 Polynomils nd Fctoring 3 Emple 4 Fctoring y Grouping Use fctoring y grouping to fctor Group terms Fctor groups is common fctor. Now try Eercise 5. STUDY TIP When grouping terms e sure to strtegiclly group terms tht hve common fctor. Fctoring trinomil cn involve quite it of tril nd error. Some of this tril nd error cn e lessened y using fctoring y grouping. The key to this method of fctoring is knowing how to rewrite the middle term. In generl, to fctor trinomil 2 c y grouping, choose fctors of the product c tht dd up to nd use these fctors to rewrite the middle term. Emple 5 Fctoring Trinomil y Grouping Use fctoring y grouping to fctor In the trinomil , 2 nd c 3, which implies tht the product c is 6. Now, ecuse 6 fctors s 6 nd 6 5, rewrite the middle term s 5 6. This produces the following Rewrite middle term. Group terms. Fctor groups. 3 is common fctor. So, the trinomil fctors s Now try Eercise 7. Activities. Completely fctor Answer: Completely fctor Answer: Guidelines for Fctoring Polynomils. Fctor out ny common fctors using the Distriutive Property. 2. Fctor ccording to one of the specil polynomil forms. 3. Fctor s 2 c m r n s. 4. Fctor y grouping.
9 33337_0P03.qp 2/27/06 9:32 AM Pge Chpter P Prerequisites P.3 Eercises See for workedout solutions to oddnumered eercises. Voculry Check Fill in the lnks.. For the polynomil n n n n... 0, the degree is nd the leding coefficient is. 2. A polynomil tht hs ll zero coefficients is clled the. 3. A polynomil with one term is clled. 4. The letters in FOIL stnd for the following. F O I L 5. If polynomil cnnot e fctored using integer coefficients, it is clled. 6. The polynomil u 2 2uv v 2 is clled. In Eercises 6, mtch the polynomil with its description. [The polynomils re leled (), (), (c), (d), (e), nd (f).] () 6 () 4 3 (c) (d) 7 3 (e) (f) A polynomil of degree zero 2. A trinomil of degree five 3. A inomil with leding coefficient 4 4. A monomil of positive degree 3 5. A trinomil with leding coefficient 4 6. A thirddegree polynomil with leding coefficient In Eercises 7 0, write polynomil tht fits the description. (There re mny correct nswers.) 7. A thirddegree polynomil with leding coefficient 2 8. A fifthdegree polynomil with leding coefficient 8 9. A fourthdegree polynomil with negtive leding coefficient 0. A thirddegree trinomil with n even leding coefficient In Eercises 6, write the polynomil in stndrd form. Then identify the degree nd leding coefficient of the polynomil In Eercises 7 20, determine whether the epression is polynomil. If so, write the polynomil in stndrd form In Eercises 2 36, perform the opertions nd write the result in stndrd form t 3 6t 3 5t w 4w w 4 9.2w y 2 4y 2 2y z 3z y 4y y y In Eercises 37 68, multiply or find the specil product y y 2y r 2 5 2r y y y t
10 33337_0P03.qp 2/27/06 9:32 AM Pge 33 Section P.3 Polynomils nd Fctoring y z 5 2z y z 3y z y y u 2 u 2 u 2 4 y y 2 y 2 In Eercises 69 74, fctor out the common fctor y z 3 6z 2 9z In Eercises 75 82, fctor the difference of two squres y z y z 5 2 In Eercises 83 90, fctor the perfect squre trinomil z 2 0z y y 6 In Eercises 9 00, fctor the sum or difference of cues y z y y 3 8z 3 In Eercises 0 4, fctor the trinomil s 2 5s t 2 t y y z z u 2 3u In Eercises 5 20, fctor y grouping In Eercises 2 52, completely fctor the epression y 2 5y 2y u 2u 2 6 u t Compound Interest After 2 yers, n investment of $500 compounded nnully t n interest rte r will yield n mount of 500 r 2. () Write this polynomil in stndrd form. () Use clcultor to evlute the polynomil for the vlues of r shown in the tle. r 2 2 % 3% 4% 4 2 % 5% 500 r 2 (c) Wht conclusion cn you mke from the tle?
11 33337_0P03.qp 2/27/06 9:33 AM Pge Chpter P Prerequisites 54. Compound Interest After 3 yers, n investment of $200 compounded nnully t n interest rte r will yield n mount of 200 r 3. () Write this polynomil in stndrd form. () Use clcultor to evlute the polynomil for the vlues of r shown in the tle. (c) Wht conclusion cn you mke from the tle? 55. Geometry An overnight shipping compny is designing closed o y cutting long the solid lines nd folding long the roken lines on the rectngulr piece of corrugted crdord shown in the figure. The length nd width of the rectngle re 45 centimeters nd 5 centimeters, respectively. Find the volume of the o in terms of. Find the volume when 3, 5, nd 7. 5 cm r 2% 3% 3 2 % 4% 4 2 % 200 r 3 45 cm 56. Geometry A tkeout fst food resturnt is constructing n open o mde y cutting squres out of the corners of piece of crdord tht is 8 centimeters y 26 centimeters (see figure). The edge of ech cutout squre is centimeters. Find the volume of the o in terms of. Find the volume when, 2, nd (45 3) 2 () Determine the polynomil tht represents the totl stopping distnce T. () Use the result of prt () to estimte the totl stopping distnce when 30, 40, nd 55. (c) Use the r grph to mke sttement out the totl stopping distnce required for incresing speeds. Distnce (in feet) Figure for 57 Rection time distnce Brking distnce Speed (in miles per hour) 58. Engineering A uniformly distriuted lod is plced on oneinchwide steel em. When the spn of the em is feet nd its depth is 6 inches, the sfe lod S (in pounds) is pproimted y S When the depth is 8 inches, the sfe lod is pproimted y S () Use the r grph to estimte the difference in the sfe lods for these two ems when the spn is 2 feet. () How does the difference in sfe lod chnge s the spn increses? cm cm 57. Stopping Distnce The stopping distnce of n utomoile is the distnce trveled during the driver s rection time plus the distnce trveled fter the rkes re pplied. In n eperiment, these distnces were mesured (in feet) when the utomoile ws trveling t speed of miles per hour on dry, level pvement, s shown in the r grph. The distnce trveled during the rection time R ws Sfe lod (in pounds) S 6inch em 8inch em Spn (in feet) R. nd the rking distnce B ws B
12 33337_0P03.qp 2/27/06 9:33 AM Pge 35 Section P.3 Polynomils nd Fctoring 35 Geometric Modeling In Eercises 59 62, mtch the fctoring formul with the correct geometric fctoring model. [The models re leled (), (), (c), nd (d).] For instnce, fctoring model for is shown in the figure. (d) () () (c) Geometric Modeling In Eercises 63 66, drw geometric fctoring model to represent the fctoriztion Geometry In Eercises 67 70, write n epression in fctored form for the re of the shded portion of the figure r r ( + 3) 4 r + 3
13 33337_0P03.qp 2/27/06 9:33 AM Pge Chpter P Prerequisites In Eercises 7 76, fctor the epression completely In Eercises 77 80, find ll vlues of for which the trinomil cn e fctored with integer coefficients In Eercises 8 84, find two integer vlues of c such tht the trinomil cn e fctored. (There re mny correct nswers.) c c c c 85. Geometry The cylindricl shell shown in the figure hs volume of V R 2 h r 2 h. () Fctor the epression for the volume. () From the result of prt (), show tht the volume is 2 (verge rdius)(thickness of the shell) h. 86. Chemicl Rection The rte of chnge of n utoctlytic chemicl rection is kq k 2, where Q is the mount of the originl sustnce, is the mount of sustnce formed, nd k is constnt of proportionlity. Fctor the epression. r R h Synthesis True or Flse? In Eercises 87 89, determine whether the sttement is true or flse. Justify your nswer. 87. The product of two inomils is lwys seconddegree polynomil. 88. The difference of two perfect squres cn e fctored s the product of conjugte pirs. 89. The sum of two perfect squres cn e fctored s the inomil sum squred. 90. Eplortion Find the degree of the product of two polynomils of degrees m nd n. 9. Eplortion Find the degree of the sum of two polynomils of degrees m nd n if m < n. 92. Writing Write prgrph eplining to clssmte why y 2 2 y Writing Write prgrph eplining to clssmte why y 2 2 y Writing Write prgrph eplining to clssmte pttern tht cn e used to cue inomil sum. Then use your pttern to cue the sum y. 95. Writing Write prgrph eplining to clssmte pttern tht cn e used to cue inomil difference. Then use your pttern to cue the difference y. 96. Writing Eplin wht is ment when it is sid tht polynomil is in fctored form. 97. Think Aout It Is 3 6 completely fctored? Eplin. 98. Error Anlysis Descrie the error Think Aout It A thirddegree polynomil nd fourthdegree polynomil re dded. () Cn the sum e fourthdegree polynomil? Eplin or give n emple. () Cn the sum e seconddegree polynomil? Eplin or give n emple. (c) Cn the sum e seventhdegree polynomil? Eplin or give n emple Think Aout It Must the sum of two seconddegree polynomils e seconddegree polynomil? If not, give n emple.
Operations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationSPECIAL PRODUCTS. The Square of a Binomial. (a b) 2 (a b)(a b) a 2 ab ab b 2. The Square of a Sum. (a b) 2 a 2 2ab b 2
4.4 Specil Products (419) 225 GETTING MORE INVOLVED 75. Explortion. Find the re of ech of the four regions shown in the figure. Wht is the totl re of the four regions? Wht does this exercise illustrte?
More informationMULTIPLYING BINOMIALS
5.4 Multiplying Binomils (527) 283 wife so tht they oth cn e expected to die in the sme yer? ) Find M(y) F(y) to get formul for the life expectncy of person orn in yer y. 2 ) 1969 ) 0.1726y 268.445 GETTING
More information5.3 MULTIPLICATION OF BINOMIALS
238 (5 16) Chpter 5 Polynomils nd Exponents 5.3 MULTIPLICATION OF BINOMIALS In this section The FOIL Method Multiplying Binomils Quickly In Section 5.2 you lerned to multiply polynomils. In this section
More informationAPPENDIX D Precalculus Review
APPENDIX D Preclculus Review SECTION D. Rel Numers nd the Rel Line Rel Numers nd the Rel Line Order nd Inequlities Asolute Vlue nd Distnce Rel Numers nd the Rel Line Rel numers cn e represented y coordinte
More informationAppendix 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 informationMULTIPLICATION OF BINOMIALS 4.3. section. The FOIL Method
4. Multipliction of Binomils (45) 98. Ptchwork. A quilt ptch cut in the shpe of tringle hs se of 5x inches nd height of.7x inches. Wht polynomil represents its re? 4.x squre inches FIGURE FOR EXERCISE
More informationChapter 9: Quadratic Equations
Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.
More informationQuadratic Equations  1
Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions  1 Sttement of Prerequisite
More informationPolynomial 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 informationFor the Final Exam, you will need to be able to:
Mth B Elementry Algebr Spring 0 Finl Em Study Guide The em is on Wednesdy, My 0 th from 7:00pm 9:0pm. You re lloed scientific clcultor nd " by 6" inde crd for notes. On your inde crd be sure to rite ny
More informationFactoring Polynomials
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
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls : The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N):  counting numers. {,,,,, } Whole Numers (W):  counting numers with 0. {0,,,,,, } Integers (I): 
More informationMultiplication and Division  Left to Right. Addition and Subtraction  Left to Right.
Order of Opertions r of Opertions Alger P lese Prenthesis  Do ll grouped opertions first. E cuse Eponents  Second M D er Multipliction nd Division  Left to Right. A unt S hniqu Addition nd Sutrction
More informationA.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
More informationa, a 1, and x is any real
MA 131 Lecture Notes Eponentil Functions, Inverse Functions, nd Logrithmic Functions Eponentil Functions We sy tht unction is n lgebric unction i it is creted by combintion o lgebric processes such s ddition,
More informationP.1 Real Numbers. Real Numbers. What you should learn
7_P.qp /7/6 9:8 AM Chpter P Pge Prerequisites P. Rel Numbers Wht you should lern Rel Numbers Rel numbers re used in everydy life to describe quntities such s ge, miles per gllon, nd popultion. Rel numbers
More informationChapter 2 Section 6 Lesson Squares, Square Roots, and Absolute Value
Chpter Section 6 Lesson Squres, Squre Roots, nd Absolute Vlue Introduction This lesson explins squred numbers, the squre root, nd the ide of bsolute vlue. Squred Numbers To squre number mens to multiply
More informationReasoning 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 informationMathematics for Chemists 2 Lecture 1: Integral calculus I. Indefinite and definite integrals, basic calculation of integrals
Mthemtics for Chemists 2 Lecture 1: Integrl clculus I Indefinite nd definite integrls, sic clcultion of integrls Johnnes Kepler University Summer semester 2012 Lecturer: Dvid Sevill Integrl clculus I 1/19
More information3 Indices and Standard Form
MEP Y9 Prctice Book A Indices nd Stndrd Form. Inde Nottion Here we revise the use of inde nottion. You will lredy be fmilir with the nottion for squres nd cubes, nd this is generlised by defining: Emple
More information4x 22x 3. Chapter 2: Variable Expressions
Chpter : Vrible Expressions Expressions (contin no sign) : An expressionis one or numbers or vribles hving some mthemticl opertions done on them. Numericl Expressions: 3 + 5 3(4) / 51 4 Expressions cn
More informationMini Lecture 8.1 Rational Expressions and Their Simplification
Mini Lecture 8. Rtionl Epressions nd Their Simplifiction Lerning Objectives:. Find numbers for which rtionl epression is undefine. Simplif rtionl epressions.. Solve pplied problems involving rtionl epressions.
More informationLinear Inequalities A linear inequality in one variable is an inequality such as
7.4 Liner Inequlities A liner inequlity in one vrile is n inequlity such s x 5 2, y 3 5, or 2k 5 10. Archimedes, one of the gretest mthemticins of ntiquity, is shown on this Itlin stmp. He ws orn in the
More informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More informationSPECIAL 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 information5.6 POSITIVE INTEGRAL EXPONENTS
54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationPrealgebra 7* In your group consider the following problems:
Prelger * Group Activit # Group Memers: In our group consider the following prolems: 1) If ever person in the room, including the techer, were to shke hnds with ever other person ectl one time, how mn
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More information131 Squares and Square Roots (Pages )
ME 3 Squres nd Squre oots (Pges 664 668) S S2.4, T MG.3, MG2., M2.7 squre root is one of two equl fctors of numer. For emple, the squre root of 25 is 5 ecuse 5 5 or 5 2 is 25. Since 5 ( 5)is lso 25, 5
More informationTHE RATIONAL NUMBERS CHAPTER
CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted
More informationSample Problems for MATH 100 Readiness Test
Montclir Stte University (MSU), Deprtment of Mthemticl Sciences Smple Problems for MATH 00 Rediness Test I. Substitution in lgebric epressions. Evlute b if = nd b =.. If y =, wht is the vlue of y when
More informationSection 6.2 The definite integral
Section 6.2 The definite integrl (3/2/8) Overview: We sw in Section 6. how the chnge of continuous function over n intervl cn be clculted from its rte of chnge if the rte of chnge is step function. We
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationThe next type of numbers we generally come to understand are fractions, but we ll put these on hold for now.
The Rel Number Line nd Types of Rel Numbers Most people s first understnding of numbers reltes to wht we mthtype people cll counting numbers or more formlly nturl numbers. These re the numbers we use
More informationALGEBRAIC PRODUCTS AND QUOTIENTS IN INDEX NOTATION
Chpter 8 Indices Contents: A Algeric products nd quotients in index nottion B Index lws C Expnsion lws D Zero nd negtive indices E Scientific nottion (Stndrd form) F Significnt figures 74 INDICES (Chpter
More informationConceptual Explanations: Exponents
Conceptul Eplntions: Eponents An eponent mens repeted multipliction. For instnce, 0 6 mens 0 0 0 0 0 0, or,000,000. You ve probbly noticed tht there is logicl progression of opertions. When you dd the
More informationRATIONAL EXPONENTS 9.2. section. calculator. closeup. Rational Exponents. Solution. Solution
78 (9 ) Chpter 9 Rdicls nd Rtionl Exponents In this Rtionl Exponents section Using the Rules of Exponents Simplifying Expressions Involving Vribles clcultor closeup You cn find the fifth root of using
More informationRational Expressions
C H A P T E R Rtionl Epressions nformtion is everywhere in the newsppers nd mgzines we red, the televisions we wtch, nd the computers we use. And I now people re tlking bout the Informtion Superhighwy,
More informationTHE CASE OF THE MISSING MIDDLE TERM EXAMPLES
THE CASE OF THE MISSING MIDDLE TERM EXAMPLES Consider the two squres tht re shown.. Wht is the re of the smller squre? Answer 2 2. Wht is the re of the lrger squre? Answer 2 3. If I took pir of scissors
More information11.2 Logarithmic Functions
.2 Logrithmic Functions In the lst section we delt with the eponentil function. One thing tht we notice from tht discussion is tht ll eponentil functions pss the horizontl line test. Tht mens tht the eponentil
More informationThe Quadratic Formula. 16 x 2 7.5x. 16 x 2 7.5x Complete the square. 16 x x x 3.75
LESSON 7.4 EXAMPLE A Solution The Qudrtic Formul Although you cn lwys use grph of qudrtic function to pproximte the xintercepts, you re often not ble to find exct solutions. This lesson will develop procedure
More informationMA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!
MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more
More informationExponential Functions
WS07.0 Eponentil Functions Aim: To study the properties of eponentil functions nd lern the fetures of their grphs Section A Activity : The Eponentil Function, f ( ).. For f ( ) : The bse of f ( ) is (ii)
More informationLinear Functions. (1) Let F denote the Fahrenheit temperature and C the Celsius temperature of an object. F and C are related by
Liner Functions A. Definition nd Exmples A function f is liner if it cn be expressed in the form f ( x) = mx + b where m nd b re constnts nd x is n rbitrry member of the domin of f. Often the reltionship
More information16 Algebra: Linear Equations
MEP Y7 Prctice Book B 16 Alger: Liner Equtions 16.1 Fundmentl Algeric kills This section looks t some fundmentl lgeric skills y emining codes nd how to use formule. Emple 1 Use this code wheel, which codes
More informationSOLUTIONS TO HOMEWORK #1, MATH 54 SECTION 001, SPRING 2012
SOLUTIONS TO HOMEWOK #, MATH 5 SECTION, SPING JASON FEGUSON. Bewre of typos. In fct, your solutions my e etter thn mine.. Some prolems hve scrtch work nd solutions. The scrtch work sys how I cme up with
More informationChapter 1. Quadratic Equations
1 Chpter 1 1.1 Eqution: An eqution is sttement of equlity etween two expression for prticulr vlues of the vrile. For exmple 5x + 6, x is the vrile (unknown) The equtions cn e divided into the following
More informationExponents. This suggests the following rule of multiplying like bases.
Eponents Objectives:. Using the product rule. Evluting epressions rised to the zero power. Using the quotient rule. Evluting epressions rised to the negtive powers. Rising power to power. Converting between
More informationChapter 6 Solving equations
Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign
More informationIntroduction to vectors
Introduction to vectors A vector is quntity tht hs oth mgnitude (or size) nd direction. Both of these properties must e given in order to specify vector completely. In this unit we descrie how to write
More information[Problem submitted by Kee Lam, LACC Professor of Mathematics. Source: Kee Lam]
Prolem ) Find the first integer of four consecutive positive integers such tht the difference of the sum of the squres of the lst three integers nd the sum of the squres of first three integers is 7. [Prolem
More informationContinuous Random Variables Class 5, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05, Spring 204 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf)
More informationSystems, Matrices, and Applications Systems of Linear Equations
Mt Dr. Firoz : System of equtions nd mtrices Systems, Mtrices, nd pplictions Systems of Liner Equtions System of eqution (Hs solution) Consistent Inconsistent (hs no solution) Dependent For Emple: Consider
More informationMA Lesson 21 Notes. **It is important to understand that a logarithm is an exponent!**
MA 15910 Lesson 1 Notes ( x5) How would person solve n eqution with vrible in n exponent, such s 9? (We cnnot rewrite this eqution esily with the sme bse.) A nottion ws developed so tht equtions such
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More information4.3 The Fundamental Theorem of Calculus
Clculus Mimus 4. The Fundmentl Theorem of Clculus We ve lerned two different rnches of clculus so fr: differentition nd integrtion. Finding slopes of tngent lines nd finding res under curves seem unrelted,
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationIntroduction. Producer s Surplus. Total Costs: Fixed and Variable
Introduction Producer s Surplus Philip A. Viton Novemer 12, 2014 We hve seen how to evlute consumer s wtp for mrketinduced price chnge: we mesure it (with pproprite cutions, primrily tht we tret it s
More information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More informationExponential and Logarithmic Functions
Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define
More informationx + 2 x x x 1 x + 1 x + 2 x + 1 x 1
2 Alger: Algeric methods 2. Rerrnging formule C HOMEWORK 2A FM A resturnt hs lrge oven tht cn cook up to 0 chickens t time. The resturnt uses the following formul for the length of time it tkes to cook
More informationTeaching Guide Zero and Negative Exponents
Teching Guide Zero nd Negtive Eponents Prepring for Your Clss Common Vocbulr Zero eponent, negtive eponent Instruction Tips In the more complicted eponentil epressions, it m be helpful to students if ou
More informationSystems of Equations
III3 Systems of Equtions Multiple Equtions with Multiple Unknowns: The generl rule tht you need to be wre of is tht to solve for two unknowns, you need two independent equtions contining those two unknowns
More informationEXPONENTS AND THEIR PROPERTIES
EXPONENTS AND THEIR PROPERTIES Bsic Definitions nd Nottion mens or 8 The number is the eponent. The number is the bse. mens is the eponent; is the bse. WARNING! The bse of n eponent is the symbol directly
More informationSections 5.2 and 5.3 Signed Numbers; Rational Numbers; Exponents; Order of Operations
Sections 5. nd 5.3 Signed Numbers; Rtionl Numbers; Exponents; Order of Opertions Number Line Negtive numbers Positive numbers Objectives 1. Perform opertions with signed nd rtionl numbers. Evlute exponentil
More informationR.2 Inequalities and Interval Notation
R Inequlities nd Intervl Nottion In order to simplify mtters we wnt to define new type of nottion for inequlities This wy we cn do wy with the more ulky set nottion This new nottion is clled using intervls
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationCommon Core Standards Addressed in this Resource
Common Core Stndrds Addressed in this Resoure NCN.9  Know the Fundmentl Theorem of Alger; show tht it is true for qudrti polnomils. Ativit pges: 34, 35 AAPR.  Know nd ppl the Reminder Theorem: For
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More informationAdding and subtracting fractions :
. Adding nd sutrcting frctions Add nd sutrct frctions y converting them to equivlent frctions with common denomintor Add nd sutrct lgeric frctions Key words frction equivlent frction denomintor common
More informationTHE ELLIPSE AND HYPERBOLA
600 (110) Chpter 11 Nonliner Sstems nd the Conic Sections 61. 1 6. 1 nd E so tht the grph of this eqution is circle. Wht does the grph of look like? B nd D cn be n rel numbers but A must equl C nd AE
More informationChapter 3 Notes Exponential and Logarithmic Functions
Chpter Notes Eponentil nd Logrithmic Functions Eponentil functions involve constnt bse nd vrible eponent. The inverse of n eponentil function is rithmic function. Eponentil nd rithmic functions re widely
More informationHonors Advanced Algebra Mr. Kellner Chapter 1 Equations and Inequalities Assignment Guide
Honors Advnced Alger Mr. Kellner Chpter 1 Equtions nd Inequlities Assignment Guide 1.1 Expressions nd Formuls Trget Gols: Use the order of opertions to evlute expressions Use formuls HW #1 pg 7 #46, 1125
More informationQuadratic Functions. Analyze and describe the characteristics of quadratic functions
Section.3  Properties of rphs of Qudrtic Functions Specific Curriculum Outcomes covered C3 Anlyze nd describe the chrcteristics of qudrtic functions C3 Solve problems involving qudrtic equtions F Anlyze
More informationQuadratic Equations. Math 99 N1 Chapter 8
Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree
More informationAssuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;
B26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndomnumer genertor supplied s stndrd with ll computer systems Stn KellyBootle,
More informationExample 1: What is the total distance traveled? What is the total displacement?
7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge
More informationChapter 2. Random Variables and Probability Distributions
Rndom Vriles nd Proility Distriutions 6 Chpter. Rndom Vriles nd Proility Distriutions.. Introduction In the previous chpter, we introduced common topics of proility. In this chpter, we trnslte those concepts
More informationQuadratic functions. Chapter12 A B. Graphs of quadratic. functions
Chpter12 Qudrtic functions Sllus reference: 2.7, 4.3, 4.6 Contents: A B C D E F G H Qudrtic functions Grphs of qudrtic functions Aes intercepts Ais of smmetr Verte Finding qudrtic from its grph Where functions
More informationnot to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions
POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the xcoordintes of the points where the grph of y = p(x) intersects the xxis.
More informationOblique Triangles and The Law of Sines Congruency and Oblique Triangles Derivation of the Law of Sines Solving SAA and ASA Triangles (Case 1)
Olique Tringles nd The Lw of Sines Congruency nd Olique Tringles Derivtion of the Lw of Sines Solving SAA nd ASA Tringles (Cse 1) Are of Tringle Lw of Sines In ny tringle ABC, with sides,, nd c, sin A
More informationSect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed twodimensionl geometric figure consisting of t lest three line segments for its
More informationUnit 4 The Definite Integral
Unit 4 The Definite Integrl We know wht n indefinite integrl is: the generl ntiderivtive of the integrnd function. There is relted (lthough in some wys vstly different) concept, the definite integrl, which
More informationSec. 1.2 Place Value and Names for Numbers
Sec. 1.2 Plce Vlue nd Nmes for Numbers Lerning Objectives: 1. Find the plce vlue of digit in whole number. 2. Write whole number in words nd in stndrd form. 3. Write whole number in expnded form. 4. Key
More informationQ u a d r a t i c E q u a t i o n s QUADRATIC EQUATIONS.
Q u d r t i E q u t i o n s QUADRATIC EQUATIONS www.mthletis.om.u Qudrti QUADRATIC Equtions EQUATIONS A qudrti eqution is n eqution where the highest inde is (squred). These re little me omplited to solve
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting
More information2 If a branch is prime, no other factors
Chpter 2 Multiples, nd primes 59 Find the prime of 50 by drwing fctor tree. b Write 50 s product of its prime. 1 Find fctor pir of the given 50 number nd begin the fctor tree (50 = 5 10). 5 10 2 If brnch
More informationLaws of Logarithms. Pre Calculus Math 40S: Explained! 213
Lws of Logrithms Pre Clculus Mth 40S: Eplined! www.mth40s.com 1 Logrithms Lesson Prt I Logrithmic to Eponentil Form converting from rithmic to eponentil form: Emple 1: Convert = y to eponentil form: Emple
More informationSECTION 64 Exact Values for Special Angles and Real Numbers
Trigonometric Functions 5. Angle of Inclintion. Recll tht (Section ) the slope of nonverticl line pssing through points P (, ) nd P (, ) is given Slope m ( )/( ). The ngle tht the line L mkes with the
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 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 informationIcebergs and Exponents Lesson 191 Basic Exponent Properties
Icebergs nd Eponents Lesson 191 Bsic Eponent Properties Lerning Trgets: Develop bsic eponent properties. Simplif epressions involving eponents. SUGGESTED LEARNING STRATEGIES: Crete Representtions, Predict
More informationhas the desired form. On the other hand, its product with z is 1. So the inverse x
First homework ssignment p. 5 Exercise. Verify tht the set of complex numers of the form x + y 2, where x nd y re rtionl, is sufield of the field of complex numers. Solution: Evidently, this set contins
More information