(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, cellOutline, groupLikeTitle, center, M18, O486, R65535, e8, 24, "Calculus"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, bold, R21845, G21845, B21845, e6, 12, "Calculus"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, R21845, G21845, B21845, e6, 12, "Calculus"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Calculus"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, B65535, 12, "Calculus"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M36, N23, O486, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M36, N23, O486, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M18, O486, l34, w351, h314, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = leftheader, inactive, M18, O486, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M18, O486, 12, "Times"; fontset = leftfooter, inactive, M18, O486, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, whiteBox, M18, O486, bold, R21845, G21845, B21845, 12, "CalcMath"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, R21845, G21845, B21845, 12, "CalcMath"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, 10, "Courier"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 10, "Courier"; paletteColors = 256; currentKernel; ] :[font = title; inactive; preserveAspect] Roots of Polynomial Equations By Kay Hall ;[s] 2:0,1;30,0;42,-1; 2:1,36,26,Calculus,0,18,65535,0,0;1,45,32,Calculus,0,24,65535,0,0; :[font = section; inactive; Cclosed; preserveAspect; startGroup] Introduction :[font = text; inactive; preserveAspect; endGroup] This project provides two programs for teachers to use when discussing the roots of polynomial equations. The first program simply graphs a series of polynomial equations varying by the constant factor,t. These graphs may be animated while various aspects of the graphs are discussed. The number of roots, number and location of real roots, and conjugate nature of imaginary roots are some of the topics that could be discussed. The second group of programs graphs the same equations beside a graph of the roots of the equations on the real-imaginary axes. This enables students to "see" more clearly the location and number of the real roots, the occurrence of double roots,and the location and the conjugate nature of imaginary roots. :[font = section; inactive; Cclosed; preserveAspect; startGroup] Graphing and Animating Polynomial Equations :[font = input; Cclosed; preserveAspect; startGroup; animationSpeed = 30] Clear[f,x] f[x_]:=x^4 - 4 x^2 + x + t Table[Plot[f[x],{x,-4,4}, PlotRange->{-10,10}], {t,-3.5,6.5}] :[font = postscript; PICT; formatAsPICT; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 217; startGroup; pictureID = 15094] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 217; pictureID = 14328] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 217; pictureID = 29061] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 217; pictureID = 17218] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 217; pictureID = 30391] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 217; pictureID = 20018] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 217; pictureID = 12488] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 217; pictureID = 12718] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 217; pictureID = 31259] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 217; pictureID = 28731] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 351; pictureHeight = 217; endGroup; endGroup; endGroup; pictureID = 30222] :[font = section; inactive; Cclosed; preserveAspect; startGroup] Graphing and Animating Polynomials and Their Roots :[font = input; preserveAspect; startGroup; animationSpeed = 30] Clear[f,x,fred,barney,a,t] f[x_] = x^4 - 4 x^2 + x + t; Do[ fred := Plot[f[x],{x,-4,4}, AxesLabel->{"x","y"}, PlotRange->{-10,10}, DisplayFunction->Identity]; (*The following line of code will store the solutions of the polynomial equations as a list of approximate numerical values stored in the variable 'sol'*); sol = N[Solve[f[x]==0,x]]; (*The following line of code does several things: first, the Re{x/.sol] will take each element of the list 'sol' and will give the real part of the number. The Im[x/.sol] gives the imaginary part of the number. If the value is a pure real number, the output of Im[x/.sol] will be 0. Using these two commands, you end up with a matrix of the solutions in a 4X2 format. The Transpose command takes that matrix and transposes it into a 2X4 format, thus allowing the graphing of the solutions as pairs of points on the real-imaginary axes.*); rts = Transpose[{Re[x/.sol],Im[x/.sol]}]; barney:=ListPlot[rts, PlotStyle->{PointSize[0.04]}, AxesLabel->{"Real","Imag"}, PlotRange->{{-2.5,2.5},{-5,5}}, DisplayFunction->Identity]; Show[GraphicsArray[{fred,barney}], Display Function->$Display Function], {t,-3.5,6.5}] ;[s] 3:0,0;224,1;383,0;1320,-1; 2:2,7,10,Courier,1,12,0,0,0;1,7,10,Courier,1,12,32767,32767,32767; :[font = postscript; PICT; formatAsPICT; output; inactive; Cclosed; preserveAspect; pictureLeft = 19; pictureTop = 12; pictureWidth = 509; pictureHeight = 149; startGroup; pictureID = 29390] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 12; pictureWidth = 509; pictureHeight = 149; pictureID = 13438] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 12; pictureWidth = 509; pictureHeight = 149; pictureID = 23403] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 12; pictureWidth = 509; pictureHeight = 149; pictureID = 6328] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 12; pictureWidth = 509; pictureHeight = 149; pictureID = 11509] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 12; pictureWidth = 509; pictureHeight = 149; pictureID = 2466] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 12; pictureWidth = 509; pictureHeight = 149; pictureID = 16522] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 12; pictureWidth = 509; pictureHeight = 149; pictureID = 309] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 12; pictureWidth = 509; pictureHeight = 149; pictureID = 914] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 12; pictureWidth = 509; pictureHeight = 149; pictureID = 20063] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 12; pictureWidth = 509; pictureHeight = 149; endGroup; endGroup; endGroup; pictureID = 28375] :[font = section; inactive; Cclosed; preserveAspect; startGroup] Some More Examples :[font = text; inactive; preserveAspect] The commands which produce the display above could easily be modified to display linear, quadratic, cubic, and quintic equations and their roots. You can modify the function line as indicated in each of the following examples. :[font = subsection; inactive; preserveAspect] Graphing Linear Equations and Their Roots :[font = input; Cclosed; preserveAspect; startGroup] Clear[f,x,fred,barney,a,t] f[x_] = x + t; Do[ fred := Plot[f[x],{x,-4,4}, AxesLabel->{"x","y"}, PlotRange->{-10,10}, DisplayFunction->Identity]; sol = N[Solve[f[x]==0,x]]; rts = Transpose[{Re[x/.sol],Im[x/.sol]}]; barney:=ListPlot[rts, PlotStyle->{PointSize[0.04]}, AxesLabel->{"Real","Imag"}, PlotRange->{{-4,4},{-5,5}}, DisplayFunction->Identity]; Show[GraphicsArray[{fred,barney}], Display Function->$Display Function], {t,-3.5,6.5,2}] :[font = postscript; PICT; formatAsPICT; output; inactive; Cclosed; preserveAspect; pictureLeft = 0; pictureTop = 1; pictureWidth = 439; pictureHeight = 128; startGroup; pictureID = 9246] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 439; pictureHeight = 128; pictureID = 10102] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 439; pictureHeight = 128; pictureID = 15384] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 439; pictureHeight = 128; pictureID = 2803] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 439; pictureHeight = 128; endGroup; endGroup; pictureID = 11560] :[font = subsection; inactive; preserveAspect] Graphing Quadratic Equations and Their Roots :[font = input; Cclosed; preserveAspect; startGroup] Clear[f,x,fred,barney,a,t] f[x_] = -4 x^2 + x + t; Do[ fred := Plot[f[x],{x,-4,4}, AxesLabel->{"x","y"}, PlotRange->{-10,10}, DisplayFunction->Identity]; sol = N[Solve[f[x]==0,x]]; rts = Transpose[{Re[x/.sol],Im[x/.sol]}]; barney:=ListPlot[rts, PlotStyle->{PointSize[0.04]}, AxesLabel->{"Real","Imag"}, PlotRange->{{-1,1},{-4,4}}, DisplayFunction->Identity, Ticks->{{-1,-.5,.5,1}, {-4,-2,2,4}}]; Show[GraphicsArray[{fred,barney}], Display Function->$Display Function], {t,-3.5,6.5,2}] :[font = postscript; PICT; formatAsPICT; output; inactive; Cclosed; preserveAspect; pictureLeft = 0; pictureWidth = 433; pictureHeight = 126; startGroup; pictureID = 5500] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 433; pictureHeight = 126; pictureID = 2843] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 433; pictureHeight = 126; pictureID = 24762] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 433; pictureHeight = 126; pictureID = 30048] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 433; pictureHeight = 126; pictureID = 19665] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 433; pictureHeight = 126; endGroup; endGroup; pictureID = 21354] :[font = subsection; inactive; preserveAspect] Graphing Cubic Equations and Their Roots :[font = input; Cclosed; preserveAspect; startGroup] Clear[f,x,fred,barney,a,t] f[x_] = x (x-2) (x + 1) + t; Do[ fred := Plot[f[x],{x,-4,4}, AxesLabel->{"x","y"}, PlotRange->{-10,10}, DisplayFunction->Identity]; sol = N[Solve[f[x]==0,x]]; rts = Transpose[{Re[x/.sol],Im[x/.sol]}]; barney:=ListPlot[rts, PlotStyle->{PointSize[0.04]}, AxesLabel->{"Real","Imag"}, PlotRange->{-5,5}, DisplayFunction->Identity]; Show[GraphicsArray[{fred,barney}], Display Function->$Display Function], {t,-3.5,6.5,2}] :[font = postscript; PICT; formatAsPICT; output; inactive; Cclosed; preserveAspect; pictureLeft = 0; pictureWidth = 433; pictureHeight = 126; startGroup; pictureID = 8934] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 433; pictureHeight = 126; pictureID = 10892] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 433; pictureHeight = 126; pictureID = 31807] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 433; pictureHeight = 126; pictureID = 7852] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 433; pictureHeight = 126; pictureID = 17273] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 433; pictureHeight = 126; endGroup; endGroup; pictureID = 31111] :[font = subsection; inactive; preserveAspect] Graphing Quintic Equations and Their Roots :[font = input; Cclosed; preserveAspect; startGroup] Clear[f,x,fred,barney,a,t] f[x_] = x (x-1) (x+1) (x-2) (x +2)+ t; Do[ fred := Plot[f[x],{x,-4,4}, AxesLabel->{"x","y"}, PlotRange->{-10,10}, DisplayFunction->Identity]; sol = N[Solve[f[x]==0,x]]; rts = Transpose[{Re[x/.sol],Im[x/.sol]}]; barney:=ListPlot[rts, PlotStyle->{PointSize[0.04]}, AxesLabel->{"Real","Imag"}, PlotRange->{{-2,2},{-5,5}}, DisplayFunction->Identity]; Show[GraphicsArray[{fred,barney}], Display Function->$Display Function], {t,-3.5,6.5}] :[font = postscript; PICT; formatAsPICT; output; inactive; Cclosed; preserveAspect; pictureLeft = 0; pictureWidth = 447; pictureHeight = 131; startGroup; pictureID = 25701] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 447; pictureHeight = 131; pictureID = 29426] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 447; pictureHeight = 131; pictureID = 10518] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 447; pictureHeight = 131; pictureID = 23470] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 447; pictureHeight = 131; pictureID = 1936] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 447; pictureHeight = 131; pictureID = 31829] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 447; pictureHeight = 131; pictureID = 3542] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 447; pictureHeight = 131; pictureID = 4111] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 447; pictureHeight = 131; pictureID = 8186] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 447; pictureHeight = 131; pictureID = 32019] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 447; pictureHeight = 131; endGroup; endGroup; endGroup; pictureID = 10616] ^*)