More about polynomialsPlace-value notation Sign-value notatione.g. Hindu-Arabic numerals12 = ********** + **e.g. oman numerals!"" = ***** + * + *#onvenient $or representing largenumbers. e.g.%&is year = 2'12Hong (ong population = )''''''"nconvenient $or representing largenumbers. e.g.%&is year = MM*""Hong (ong population = +++#onvenient $or mat&ematicaloperations"nconvenient $or mat&ematicaloperationsXXXIXII???Arithmetic Algebra"ntegers ,in place-value notation-.2 1 '2 1 '1' ... 1' 1' 1'nnd d d d + + + +/&ere' 1 20 0 0...0 1'010 20...0 23nd d d d Polynomials /it& 1 un4no/n11 1 '...n nn na x a x a x a+ + + +/&ere' 1 20 0 0...0 1... 20 10 '010 20...3na a a a 56ample.7 2 1 '12781 1' 2 1' 7 1' 8 1' = + + + 56ample.7 27 2 1 '2 7 81 2 7 8x x xx x x x+ + += + + + #orrespondence.unit digit 9 constant termtens digit 9 coe$$icient o$ x&undreds digit 9 coe$$icient o$ x21' 9 x1'2 9 x2etc.:&at do you see+12 + 78;eginners see 8 digits.Advanced students see 2 integers.,2x + 1- + ,x < 2-;eginners see 8 terms.Advanced students see 2 polynomials.#onsider a polynomial as a /&ole0 instead o$ a collection o$ parts."t is =ust li4e &o/ you perceive integers instead o$ digits.1More about polynomialsSimilaritiesAddition72 + 21 = >7Addition,7 2- ,2 1- > 7 x x x + + + = +Subtraction72 < 21 = 11Subtraction,7 2- ,2 1- 1 x x x + + = +Multiplication Multiplication22,7 2-,2 1-,7 2- ,? 8 -? ) 2x xx x xx x+ += + + += + +Ma=or di$$erence#arrying2@ + 7> = ?7Ao carrying,2 @- ,7 >- > 17 x x x + + + = +not ?x + 7+ < B bet/een t/o integers%&e ans/er is also an integer.2 + 7 = >2 < 7 = -12 B 7= ?+ < B bet/een t/o polynomials%&e ans/er is also a polynomial. ,2x + 1- + ,x < 2- = 7x < 1,2x + 1- < ,x < 2- = x + 7,2x+1- ,x0 )0 110 170 Q2 = 1B2#omposite numbers80 ?0 @0 20 1'0 120 Q8 = 1B8 = 2B2"rreducible Polynomials ,%/o types-Minear ,x up to po/er 1- $actorse.g. ,x + 1-0 ,2x < )-Huadratic ,x up to po/er 2- $actors2, - ax bx c + + 0 /&ere 28 b ac is not aper$ect sFuare. e.g.2, > 1- x x + +Huestion. Ho/ can /e 4no/ t&at an integer can uniFuely be represented as a product o$ primes+e.g. #an /e &avesome integer = 2B) = 7B>,%&e result is not clear /&en t&e numbers are very large.-Huestion. Ho/ can /e 4no/ t&at a polynomial can uniFuely be represented as a product o$ irreducible $actors+Ans/er. Nundamental %&eorem o$ Arit&meticAny integer can be uniFuely e6pressedas a product o$ primes.,5uclid ;# 7'' et al-"n ot&er /ords. %&e $actoriLation o$ aninteger &as only one ans/er.Ans/er. Nundamental %&eorem o$ Algebra ,Sort o$-Any polynomial can be uniFuely e6pressed as a product o$ irreducible $actors.,#. N. Kauss 1)22-"n ot&er /ords. %&e $actoriLation o$ an integer &as only one ans/er.Ho/+ S&ort division2 12'2 ?'2 7'7 1>>12' = 2B2B2B7B>Ho/+ational oot %&eoremNactor %&eoremMong Givision#ross Product,as s&o/n belo/-Nactor %&eorem"$' = nPm0 t&en ,mx < n- is a $actor o$ P,x-.Nrom emainder %&eorem. P,x- = ,mx < n-BHuotient + emainder."$ emainder = '0 P,x- = ,mx < n- B Huotient. i.e. ,mx < n- is a $actor o$ P,x-."t &appens /&en' = nPm.4More about polynomials56ample. NactoriLe 7 27 18 ) 8 x x x + + PlanningNactor %&eorem tells us t&at i$ /e can $ind nmso t&at'nPm = 0 /e can &ave, - O , - mx n Px . ;ut /&at s&ould /e c&oose as nm+ational oot %&eorem"$'nPm = 0 t&en m O t&e leading coe$$icient ,t&e $irst number-0 andn O t&e constant term ,t&e last number-. Nrom 7 2, - 7 18 ) 8 Px x x x = + + Meading coe$$icient= 7#onstant term = -8Possible mDs. R10 R7Possible nDs.R10 R20 R8%ry 1 2 810 20 80 0 07 7 7nm = Sou t&in4Q Sou /riteQMet7 2, - 7 18 ) 8 Px x x x = + + MetDs Start /it& t&e easy one.7 2,1- 7 1 18 1 ) 1 8 ' P = + + :&oops.%ry again7 2, 1- , 1- , 1- 1', 1- @ ' P = =SayJ :rite do/n only t&e success$ul one.7 2, 1-, 1- , 1- 1', 1- @'P = = ,x + 1- is a $actor o$ P,x-.Mong division,#orrect calculation gives ' as t&e remainder.-#ross product.GoneJ7 227 18 ) 8, 1-,7 11 8-, 1-, 8-,7 1-x x xx x xx x x + + = + + = + + %&e calculator may &elp. Programs are available at&ttp.TT/ongsircalculator.org$ree.comT5nter t&e cubic $ormula ,-. %&e roots are readily available.5More about polynomialsSolution #oncept 7. 56tend our set o$ number T algebraic e6pression to includet&e ans/ers o$ division."ntegers. 1Q0 -20 -10 '0 10 20 Q3Polynomials11 1 '...n nn na x a x a x a+ + + +56tend 56tendational numberspqp0 q are integers0 q'ational e6pressions, -, -PxQxP,x-0 Q,x- are polynomials0 Q,x- '56amples. 7 2220 20 '.>08 1'' 56amples. 22210 18 8xx xx x + +"ntegers. 18 E 8 = + ational numbers. )18 82 = Polynomials. 2, 1- , 1- + x x = ational 56pressions.21, 1- , 1-1x xx =+ Multiplication and Givision2 71' 7>22 22 1 18 2 ) ?x x xx x x+ + + + +2 7>1' 7= to eciprocal2 222 1 2 ) ?8 1x x x xx x+ + + += +27 > )2 > 7= NactoriLation2, 1- , 2-,2 7-, 2-, 2- 1x x xx x x+ + += + +7 )2 1= #ancellation1 2 72 1x xx + += 7 )2 1=Aumerators multiply0denominators multiply., 1-,2 7-, 2- 1x xx+ += 212= Ans/er22 > 72x xx+ +=Addition and Subtraction) >12' 7?+2 2> 1@ @ 8 @ 8xx x x+ + +) >12 1' 12 7= + 56press t&edenominator as amultiple o$ t&e K.#.G.,as s&o/n belo/->8, 1- 2, 1-18, 1- , 1-x xxx x=+ ++ +) 7 > 1'12 1' 7 12 7 1' = + Nind t&e commondenominator. Ae/denominator = M.#.M.o$ old denominators.,as s&o/n belo/-> , 1-8, 1- 2, 1- , 1-, 1- 2, 1-8, 1- , 1- 2, 1-xx x xx xx x x +=+ + ++ + ) 7 > 1'12 1' 7 + = Same denominator0add or subtractnumerators directly22>, 1- 2, 1-@, 1- , 1-x xx x+ + =+ )17?'=Ans/er222 )@, 1- , 1-x xx x+ +=+ Ho/ to $ind t&e K.#.G. and M.#.M+K.#.G.%a4e t&e lo/est po/er o$ eac& primeT irreducible $actorM.#.M.%a4e t&e &ig&est po/er o$ eac& primeT irreducible $actor7 1 12 2 '2 1 '7 2 112' 2 7 >7? 2 7 >K.#.G.=2 7 > 12M.#.M.=2 7 > 7?'= = = =2 7 1 12 2 2 '2 1 '7 2 1 2@ @ 2 , 1- , 1-8 @ 8 2 , 1- , 1-K.#.G.=2 , 1- , 1- 8, 1-M.#.M.=2 , 1- , 1- @, 1- , 1-x x xx x x xx x xx x x x = + + + = + + = + + = + 6More about polynomials5Fuations vs. Polynomials5Fuations Polynomials56ample.1 2 x + = 56ample.1 x +Psual goal. solve $or x Psual goal. simpli$y:&en x = 10 M.H.S. = .H.S.%&ere$ore x = 1 is a solution.:&en x = >0 M.H.S. U .H.S.%&ere$ore x = > is not a solution.Aot simpli$ied., 1- ,2 7- x x + + +Simpli$ied. 7 8 x +:&at can be done.+ < B E t&e same number on bot& sides.:&at can be done.56pand0 $actoriLe0 combine li4e terms.Gomain o$ x.Psually real numbers.Gomain o$ x.GoesnDt matter. :e are not $inding x.Solving an 5Fuation by Polynomials5Fuations Polynomials56ample. Solve7 27 18 ) 8 ' x x x + + =#onsider t&e polynomial7 2, - 7 18 ) 8 Px x x x = + + Gi$$icultJ NactoriLe, 1-, 8-,7 1- ' x x x + + =, - , 1-, 8-,7 1- Px x x x = + + 110 8 or 7x = C%&is =D is not eFual to C%&at =D= $or values = $or polynomialsHolds /&en t&e values o$ bot& sidesare t&e same.Holds /&en t&e correspondingcoe$$icients and constant terms onbot& sides are t&e same.56ample 1. #omparing x + 1 and 2%&e value o$ x + 1. varies /it& x%&e value o$ 2. $i6edM.H.S.Polynomial Cx+1D#oe$. o$ x = 1#onst. term = 1.H.S.Polynomial C2D#oe$. o$ x = '#onst. term = 2x + 1 = 2Holds only /&en x = 1.x + 1 2Gi$$erent polynomials56ample 2. #omparing ,x + 1- ,x < 1- and x2 < 1Nor any real x0,x + 1- ,x < 1- and x2 < 1gives t&e same value.M.H.S.PolynomialC,x + 1- ,x < 1-DA$ter e6pansion#oe$. o$ x2 = 1#oe$. o$ x = '#onst. term = 1.H.S.PolynomialCx2 < 1D#oe$. o$ x2 = 1#oe$. o$ x = '#onst. %erm = 1,x + 1- ,x < 1- x2 < 1"dentity,x + 1- ,x < 1- = x2 < 15Fual T eFuivalent polynomials7