00020002010015ExplorePolynomial.ACT0008020011DescartesRule.EAC010000001b4a DescartesRule.EACExplorePolynomial.ACT/3  $ E '*[4$o p of Signs (Beautiful!),/How many real sol)o4does a p have?\kk-->톸|1The number of positive real roots is equto t. alteration&nsignIf() or less thaJk by a fact"wtwo.  negj-⛏-Positive real roots? 7kf(x) has 3 alterations in sign. This implies there are.pV or just 1. {NegnQf(-1m{be~nZ~ X+will!One<, one and 4 imaginary .\  - ORIethregeal roots and 2 imaginary .  #"So, which is it? Check the graph! 2[;\C Answer for Q2!08Positive real roots? 7vf(x) has 4 alterations in sign. This implies there are.pV, 2 or no~. NegyTf(-1exactly4n]%%sibil*es:;andM 2 imaginary - OR -n;(W+and 4 imaginary roots.  - OR -  ;0 positive =1 negarealA6V es[bJus/ thought...3Do or complex always e in pairs?eAct020013EvenDegreePolys.EAC0100000003c3 =^(4)-3^(2)+2  `G'a&`#Y @v$Qr ^A 'L'pC uxy q@# u x^4-3 2+2++ EvenDegreePolys.EACExplor nomial.ACT15  $ E '[4%Zing ]s of  s-Άl]y*In the linked equation above, try changing(ead $coefficient to - and press EXE. Also?degree2anox-r positive even integer.[ Complete:[ For =an+bn-1+..., SR 7if n is a{ AND a!hen? as ,____-vnegvas -, ____[*If f() is an even degree polynomial, thits ends will always go in  _ direction. The ofE@ ing behaviorw determined byj sig1_.eAct020015ExploreLinearEqns.EAC0100000002ec ExploreLinearEqns.EACPolynomial.ACT25 BF $kE ('[s(sing the Slope of a Equation0p*-Intercept form9l:/ y=m+bE\}d ns Tap-->Έ"݌$$f5D `SE uX xc3qR D`v2.r mA 'L' ux'y !`@ 6'Bq2 YG!5I   CK[ [ ]o{ y=2x-2%"Experiment with the value of m in$link above by changing 2 to a differC: and press$ EXE. Then'scusscquestions below.D,#aC meanxa positislopeBk#b3nega33;c. The meaning of a slope when it is less than one.[6dCgreaterFezeroufn+eAct020017Explore_#_Solutions.EAC01000000031f Explore_#_Solutions.EACPolynomial.ACT59  $ E '[4uing uMHow many positive and nega real number s does each equ) on have:[V1. d2l -3+2=0'2'+'3. 2O7+3O4. -w+5+4w5+4+55EHint:Drag and drop left side of equation to graph it.\Tap hereopen->Έ#4 `G'a&`KY.4 Dpv$`r mA 'L'C uxy v [ Discuss:bCan we determine the numberDpositivecnegQ real%solufs by exaEingGsigns?zS[Consider replacing with -. What information can you obtain from the signs now?[TDescartes was a greTOheTcian! To learn more, seeV<Rule eActivity.deAct020017Explore_a(-h)^2+k.EAC010000000232 y=-3(x-2)^(2)+1   ` YTBh@# Y @vΆ (What dooordinates (h,k) rem?eAct020017Explore_a^2+b+c.EAC010000000281 y=2x^(2)+2x-2  ` D?@v"Qr ^A 'L'pC uxy qq@  2x^2+ -2, Explore_a^2+b+c.EACPolynomial.ACT59  $ E ' [&u:[= 2(P" in the equation below by changingcoefficients and4n press#EXE:]\First, tap here -->Ά8What happens to graph as wecrease value of a?H0HifH make a negve@2'How does changing the value of c effectgraph?eAct020014FactorWithGraphs.EAC0100000003fd FactorWithGraphs.EACExplorePolynomial.ACT26  $ E '[4 "Using a p to Quadratics*[  : 2#-4+3\1TapQsee g->ΆO  `G'a&`Y.4 Dv`r mA 'L' ^ uxy q@   x^2-4 +3  [ Now, factor:[ 2(-4+3=( ?))CK&ΈC6 `G'a&`kY.4 DrvQr ^AL C uxy   N@  N x^2+-12''x x[Bonus'Using the graph, guessfactors of: [FP3-52n+4=?\| Tap to seh->ΈΊ܊7 `G'a&`RY D?vQri^A Ϛx y   'B  ' x^3-5 2+4 00Z Z[hoeAct020012OddDegreePolys.EAC010000000392 =^(3)+^(2)-3+1  `G'a&`#Y  @v$Qr ^A 'L'pC uxy q@# ux^3+2-3+1//% OddDegreePolys.EACExplor nomial.ACT04  $ E '[4$Zing ]s of  s,;What is the long-run behavior<pKodd dK?K\Tap Here ---->Άk]x*In the linked equation above, try changing(ead coefficient to neg8ve and press EXE. AlsoFdegree9anor posiAodd integer.[ Complete:[ For =a-n+bn-1=+..., if n is az AND a hen as ,~____-6vOvuas -, ____[GZIf f() is an odd degree polynomial, then its ends will always go in Z^ directions.jeAct010008main.ACT0001020012eActivity Save.EAC010000000392 =^(3)+^(2)-3+1  `G'a&`#Y  @v$Qr ^A 'L'pC uxy q@# ux^3+2-3+1//% OddDegreePolys.EACExplor nomial.ACT04  $ E '[4$Zing ]s of  s,;What is the long-run behavior<pKodd dK?K\Tap Here ---->Άk]x*In the linked equation above, try changing(ead coefficient to neg8ve and press EXE. 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