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


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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.
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