Math 111! Name:___!Key!_______________________________________________________________Grp#_______!GrpAct – Week 5A!! ! ! ! ! Polynomials!2! ! ! !!!!!Sections!4.2,!4.3,!&!4.4!!Prerequisite!Skills!Key!Terms!Learning!Objectives!• factoring!• distributing!polynomials!• Interpretin g!p o ly n o mials!and!sketching!them!based!off!of!the!function!• Finding!roots!of!a!polynomial!!• Solving!in!a!piece-wise!function!§ polynomial!§ leading!coefficien t!§ degree!§ root/zero/x-intercep t!§ multiplicity!!§ factor!!§ factored!form! !§ end!behavior!§ continuous!!-!Model!with!polynomial!function!-!Find!the!roots!of!polynomials!in!factored!form!-!Sketch!graphs!of!polynomials!in!factored!form!-!Identify !degree,!multiplicity!of!polynomial!roots,!and!end!behavior!in!factored!fo rm !an d!in !grap h ical!fo rm!-!Construct!formulas!for!polynomials!given!in!graphical!form!-!Solve!polynomial!inequalities!-!Determine!if!a!piecewise!polynomial!function!is!continuous!on!its!doma in .!-!Understand!how!end!behavior!depends!only!on!the!leading!term!!1. Do!not!use!a!calculator!for!any!part!of!this!problem.!The!polynomial!function!f!is!given !in !fa ct or ed !fo rm!here:! !f (x) = −23x x − 5( )3x − 1( )x + 3( )2!a. What!is!the!degree!of!f?!Hint:!You!do!not!need!to!multiply!the!function!out!to!answer!this,!but!you!can!still!think'about!what!the!highest!power!of!x!would!be!if!it!were!multiplied!out.! 7 b. Describe!the!end!behavior!of!f.!That!is,!expla in !w h at!h a p pe n s!to !f(x)!as!x! ap proaches!infinity,!and!as!x!approaches!negative!infinity.!!!as x → ∞, f (x)→ − ∞as x → − ∞, f (x)→ ∞!!!!c. List!the!roots!of!f,!an d !s ta te !t h e !multiplic ity !o f !e ac h .!Zero!!!!!Multip!! ! 0! 1! !! ! 5! 3!! ! 1! 1!! !!!!!!!!!!!!–!3! 2!!d. WITHOUT!USING!A!CALCULATOR,!sketch!a!!possible!graph!of!f.!!!!!!!e. Solve!the!inequality!f (x) < 0!without!using!a!calculator.!(!3,!∞)!!!!!!!2. Explain!how!to!determine!the!degree!of!a!polynomial!from:!!a) its!factore d !fo rm.! !The!degree!of!a!polynomial!can!only!be!determined!when!it!is!in!standard!form!so!we!must!determining!the!leading!term!of!the!polynomial!in!standard!form!by!thinking!about!multiplying!it!out.!!b) Can!you!determine!the!exact!degree!of!a!polynomial!from!its!graphical!representation?!!!YES! NO c) Why!or!Why!Not?!Explain! We!can!only!determine!the!Minimal!Possible!degree.!!Look!at!the!graph!of!!!!!!!!f (x) = x3and g(x)= x5!!in!the!same!wind o w .!!If!g iv en !ju s t!o n e!o f!t h es e !fu n ctio n s !gr ap h ic ally !w o u ld !w e !know!if!it!was!3rd!degree!or!5th!degree?!!3. Solve!the!inequality!x − 2( )x + 3( )x + 1( )2≥ 0!!−∞,−3(⎤⎦∪{−1}∪ 2,∞⎡⎣)!!!4. A!formula!for!a!piecewise-defined!function !is!give n!be low .! !f (x) =12x + 2 if − 3 ≤ x < −1−x2+ 2 if − 1 ≤ x ≤ 2⎧⎨⎪⎩⎪!1) What!is!the!domain!of!f?! !!−3,2⎡⎣⎤⎦!!!2) Evaluate!Each!of!the!following:!!f (−5)= ? -5!is!not!in!the!domain!so!this!is!undefinedf (−3)=12f (−1)= 1f (1) = 1!!!3) Solve!the!equation!f(x)!=!0!! The!only!zero!is!!!x = + 2!!!4) Sketch!a!graph!of!f!on!the!axes!at!right.! ! ! ! !!5) Is!f!continuous!on!its!domain?!Explain!wh y!or!why!not.!f!is!NO T !c o n tin u o us !on!its!domain!since!it!has!a!jump!at!x'='-1!!!!!!Explain!how!to!solve!a!polynomial!inequality.!!!!What!does!it!mean!for!a!function!to!be!continuous?!!!!!6) Four'polynomial'functions'are'graphed'below.'For'each'you'will'(a)'determine'if'the'degree'of'the'polynomial'is'odd'or'even;'(b)'determine'if'the'leading'coefficient'is'positive'or'negative;'(c)'identify'the'roots'and'their'multiplicities.'A. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!B.!! ! !!!!!!!!!!!!! !! ! Degree!(circle!one):!! ! Odd! Even! ! ! Degree!(circle!one):!! Odd! Even! !Leading!Coefficient!(circle!one):!! Positive!!!!Negative! Leading!Coefficient!(circle!one):!! Positive!!!!Negative!List!each!root!and!its!multiplicity:! ! ! List!each!root!and!its!multiplicity:!! !!!! ! C.!! ! ! ! ! ! ! ! D.!! !!!!!!!!!!!!! !! ! !!!!!!! ! Degree!(circle!one):!! ! Odd! Even! ! Degree!(circle!one):!! ! Odd! Even! !Leading!Coefficient!(circle!one):!! Positive!!!!Negative! Leading!Coefficient!(circle!one):!! Positive!!!!Negative!List!each!root!and!its!multiplicity:! ! ! List!each!root!and!its!multiplicity:!!Match!each!function!above!with!its!formula!below,!and!indicate!if!the!leading!coefficient!a!shou ld!be!positive!or!negative.!Note!that!there!is!not!enough!information!given!to!determine!the!specific!values!for'a.!Function!Graph!!(A,!B,!C!or!D)!a!<!0!!or!!a!>!0!!Function!Graph!(A,!B,!C!or!D)!a!<!0!!or!!a!>!0!2() ( 1)( 1)fx ax x=+ -!A!a!<!0!!22() ( 1)( 1)hx axx x=+-!C!a!>!0!2() ( 1)( 1)kx ax x x=+-!D!a!<!0!!22() ( 1)( 1)gx ax x=+ -!B!a!>!0!-2-112-2-112Root!Multiplicity!!Root!Multiplicity!-1!2!!-1!2!1!1!!1!2!!!!!!Root!Multiplicity!!Root!Multiplicity!-1!0!1!2!1!2!!-1!0!1!1!2!1!-2-112-2-112!!!-5-4-3-2-112-4-2246810What!is!the!degree!of!g?!!Explain!how!to!determine!the!degree!of!a!polynomial!from!its!graph.!!7) A!polynomial!function!g!is!graph e d !b elo w .!Give!a!formula!for!g(x)!with!low e st!po ss ible!d egre e.!In!a dd ition !to!identifyin g !th e!r o ot s,!a n d !th eir !m u lt ip licitie s ,!yo u !w ill!n e ed !t o !ide n tify !th e !le ad in g !co e ffic ien t !a.!!To!do!this ,!yo u !can !use!the!fact!that!the!point!(-3,!2)!lies!on!the!graph!of!g.!!!!!!!!!!! !!!g(x) = −12(x + 4)(x + 2)2(x − 1)!8) Lets!compare!the!growth!rate!of!polynomials.!!For!example,!g(x)!=!x6!and!h(x)!=!x4!have!the!same!end!behavior!(for!both!functions!the!graph!rises!to!the!left!and!the!right),!but!does!one!grow!faster!than!the!other?!!!!a. Consider!the!degree!5!polynomialf (x) = 2x5+ 3x4+ 4x3+ 2x2+ 6x.!We!wa n t!to !inv es tiga te !h ow!this!polynomial!grows!as!x!becomes!large!and!positive.!!Please!fill!in!the!last!column!of!the!table.!x(2x5!3x4!4x3!2x2!6x(f(x)!=!!2x5+3x4+4x3+2x2+6x(%(of(f(x)(that(comes(from(the(leading(term:(!1!2!3!4!2!6!17!2/17=0.117!11.7%!10!200,000!30,000!4000!200!60!234,260!200000/234260!85.4%!100!20,000,000,000!300,000,000!4,000,000!20,000!600!20,304,020,600!98.5%!1000!2,000,000,000,000,000!3,000,000,000,000!4,000,000,000!2,000,000!6000!2,003,004,002,006,000!99.85%!!As!x!becomes!large,!you!should!see!that!the!highest!powered!term!becomes!much!more!significant!than!the!other!terms.!We!sa y!tha t!the!h ighe st!po we red !term !dominates!the!po lyno m