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render = function render(){var _vm=this,_c=_vm._self._c;return _c('div',[_c('h3',{ref:\"intro\"},[_vm._v(\"\\n    Zero of a Polynomial\\n  \")]),_c('p',[_vm._v(\"\\n    Zeros of a polynomial, say \\\\(p(x)\\\\),  are the values of variable \\\\(x\\\\) at which\\n    the polynomial becomes zero.\\n    Thus, if \\\\(x=x_0\\\\) is a zero of the polynomial \\\\(p(x)\\\\) then it must satisfy the following:\\n    $$p(x=x_0) = 0$$\\n    The zeros of a polynomial are sometimes also called its roots.\\n    Depending on the degree of the polynomial, it can have one or more than one zeros —\\n    as we will discuss in the following section.\\n  \")]),_c('h3',{ref:\"formal\"},[_vm._v(\"\\n    Number of Zeros of a Polynomial\\n  \")]),_vm._m(0),_c('h3',{ref:\"playgraph\"},[_vm._v(\"\\n    MagicGraph | Zeros of a Polynomial\\n  \")]),_c('p',[_vm._v(\"\\n    Through this visually interactive MagicGraph, students will learn about the\\n    zeros of the polynomial, and how to calculate them using the graphical method.\\n  \")]),_c('h5',[_vm._v(\" To Get Started: \")]),_vm._m(1),_c('h5',[_vm._v(\" To Explore:\")]),_c('p',[_vm._v(\"\\n    After finding the zeros of one polynomial function, tap on the shuffle button to create a new polynomial function\\n    Each time you tap on the shuffle button,\\n    the MagicGraph generates a new polynomial for you to practice with.\\n    Using the graphical method described above, find the zeros of the polynomial — practice as many times as you want.\\n  \")]),_c('v-responsive',[_c('v-layout',{attrs:{\"justify-center\":\"\"}},[_c('div',{staticClass:\"edliy-box-about\",attrs:{\"id\":\"jxgbox1\"}})])],1)],1)\n}\nvar staticRenderFns = [function (){var _vm=this,_c=_vm._self._c;return _c('p',[_vm._v(\"\\n    The fundamental theorem of algebra states that a non-zero polynomial has\\n    the same number of zeros (or roots) as its degree.\\n    This requires that we account for multiplicity of zeros (i.e. repeated zeros).\\n    Thus, a polynomial of degree \\\\(n\\\\) will have \\\\(n\\\\) zeros or roots. Some of these zeros may be repeated or complex.\\n    Below are few specific examples.\\n    \"),_c('ul',{staticClass:\"a\"},[_c('li',[_c('h5',[_vm._v(\" Linear Polynomial (Degree =1) \")])]),_vm._v(\"\\n      A linear polynomial (i.e. a polynomial of degree 1) will have one zero.\\n      To give an example, let's consider the linear polynomial $$p(x) = 2x-6$$\\n      This polynomial has exactly one zero, which is \\\\(x=3\\\\).\\n      \"),_c('li',[_c('h5',[_vm._v(\"Quadratic Polynomial (Degree =2)\")])]),_vm._v(\"\\n      A quadratic polynomial (i.e. a polynomial of degree 2) will have two zeros\\n      (note that sometimes these zeros may be repeated).\\n      Let's first discuss the example of a quadratic polynomial that has two distinct zeros.\\n      $$q(x) = x^2 -6x +8$$\\n      After a little rearrangement, \\\\(q(x)\\\\) can be rewritten as:\\n      $$q(x) = (x-4)(x-2)$$\\n      You can see that this polynomial becomes zero at \\\\(x=2\\\\) and at \\\\(x=4\\\\). These are the two zeros of this polynomial.\\n      \"),_c('br'),_vm._v(\" Next, we discuss the example of a quadratic polynomial that has two repeated (or identical) zeros.\\n      Consider the following polynomial:\\n      $$r(x) = x^2 -4x +4$$\\n      This polynomial can be rearranged as follows:\\n      $$r(x) = (x-2)(x-2)$$\\n      Thus, this polynomial has two repeated zeros at \\\\(x =2\\\\).\\n    \")])])\n},function (){var _vm=this,_c=_vm._self._c;return _c('p',[_vm._v(\"\\n    You are given a polynomial function named f(x).\\n    On this MagicGraph, you will also see the graph representing this polynomial function.\\n    You goal is to identify the zeros (or roots) of this polynomial using the graphical method (described below).\"),_c('br'),_vm._v(\"\\n    Point 'P' is a movable point that can glide along the polynomial curve.\\n    As you drag this point along the polynomial curve, the MagicGraph will\\n    automatically calculate and display the value of the polynomial at point P.\\n    Identify the location (or locations) of point P on the graph at which the polynomial becomes zero.\\n    These are going to be the zeros (or roots) of the polynomial function f(x).\\n  \")])\n}]\n\nexport { render, staticRenderFns }","\r\nconst Boxes = {\r\n  box1: function () {\r\n   JXG.Options.board.minimizeReflow = 'none';\r\n\t var brd2 = JXG.JSXGraph.initBoard('jxgbox1',{boundingbox: [-10, 14, 10, -6],\r\n    keepaspectratio: true, axis:true, ticks:{visible:false},\r\n    grid:true, showCopyright:false, showNavigation:false,\r\n    pan:{enabled:false}, zoom:{enabled:false}});\r\nbrd2.suspendUpdate();\r\nvar resize1 = function () {\r\n  brd2.resizeContainer(brd2.containerObj.clientWidth, brd2.containerObj.clientHeight, true);\r\n  brd2.fullUpdate();};\r\nwindow.addEventListener(\"resize\", resize1);\r\nbrd2.options.layer['image'] =14;\r\nvar vis=0;\r\n\r\n//Title and subtitle\r\nbrd2.create('text', [0, 13, 'Zeros of a Polynomial'],{display:'internal', highlight:false, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald;fontWeight:bold',fontSize:function(){return Math.round(32*brd2.canvasWidth/800.)}},);\r\nbrd2.create('text', [0, 12, '(Part I: Graphical Illustration)'],{visible:()=>vis==0,highlight:false, display:'internal', fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(26*brd2.canvasWidth/800.)}},);\r\nbrd2.create('text', [0, 12, '(Part II: Direct Verification)'],{visible:()=>vis==1,highlight:false, display:'internal', fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(26*brd2.canvasWidth/800.)}},);\r\n\r\n//Variables\r\nvar qa = [0, 0, 4, 9, -2, 9, -7, 4, 8, -7, 4, 0, 0, 0, 0 , 0];\r\nvar qb = [1, 2, -2, -3, 4, 2, -7, 8, -8, 0, -5, 5, 3, -3, -5, 2];\r\nvar qc = [1, -3, -2, 0, 4, -5, 1, -4, -5, 5, -4, 2, 5 , 1, 2, -3];\r\nvar n = 5;\r\nvar a1 = 1;\r\nvar b1 = -3;\r\nvar c1 = 1;\r\nvar o1 = 2;\r\nvar txta1 = '1 x^2';\r\nvar txtb1 = '- 3 x ';\r\nvar txtc1 = '+ 1';\r\nvar r1 = 0;\r\nvar r2 = 0;\r\n//var roots = [0];\r\n\r\n//Shuffle image\r\nvar shuffle1 = brd2.create('image', ['/assets/shuffle.svg', [-0.75, -5], [1.5, 1.5]], {visible: true, fixed: true});\r\nshuffle1.setLabel('Tap to Shuffle')\r\nshuffle1.label.setAttribute({visible:false, offset:[15, -15], CssStyle:'fontFamily:Oswald', fontSize:16});\r\nshuffle1.on('over', function () {this.label.setAttribute({visible:true});});\r\nshuffle1.on('out', function () {this.label.setAttribute({visible:false});});\r\n\r\n//Shuffle function\r\nshuffle1.on('down',function(){\r\n\r\n    x1txt.setAttribute({visible: false});\r\n    x2txt.setAttribute({visible: false});\r\n    root1Point.setAttribute({visible: false});\r\n    root2Point.setAttribute({visible: false});\r\n    pDragMe.moveTo([-10,5], 10);\r\n\r\n\r\n    o1=0;\r\n    while(o1 < 2){\r\n        o1 = Math.round(Math.random()*2);\r\n    }\r\n\r\n        n = Math.floor(Math.random()*qa.length);\r\n\r\n        a1 = qa[n];\r\n        b1 = qb[n];\r\n        c1 = qc[n];\r\n\r\n        if(a1== 0){\r\n            txta1 = ' ';\r\n\r\n        }\r\n        else if(a1 != 0){\r\n            txta1 = a1 + ' x^' + o1 + ' ';\r\n        }\r\n\r\n        //b1 = Math.round(Math.random()*20-10);\r\n\r\n        if(b1== 0){\r\n            txtb1 = ' ';\r\n        }\r\n        else if(b1 > 0){\r\n            txtb1 = '+ ' + b1 + ' x ';\r\n        }\r\n\r\n        else{\r\n            txtb1 = ' ' + b1 + ' x ';\r\n        }\r\n\r\n        //c1 = Math.round(Math.random()*10-5);\r\n\r\n        if(c1 == 0){\r\n            txtc1 = ' ';\r\n        }\r\n        else if(c1>= 0){\r\n            txtc1 = '+' + c1;\r\n        }\r\n        else{\r\n            txtc1 = c1;\r\n        }\r\n/*\r\n    var equation = a1 + '*x^' + o1  + txtb1 + '*x' + txtc1 + '=0';\r\n    //console.log('equation:',equation)\r\n    roots = nerdamer.solveEquations(equation,'x');\r\n    var num_roots=roots.length;\r\n    //console.log('num roots=',roots.length)\r\n    for(var r=0;r<num_roots;r++){\r\n        console.log('roots',r,nerdamer(roots[r].totxting()).evaluate().text());\r\n    }\r\n    //console.log('roots2',roots.totxting());\r\n*/\r\n});\r\n\r\n//GRAPH\r\nvar graph = brd2.create('functiongraph', [function(x){ return a1*Math.pow(x,o1)+b1*x + c1;}, -20, 20],{strokeColor:'grey', strokeWidth:2});\r\n\r\n//First root\r\nvar axisX = brd2.create('line', [[-5, 0], [5,0]], {visible: false, fixed: true});\r\nvar zero1 =brd2.create('intersection',[axisX, graph, 0],{visible: false, size:4, name:'Zero', withLabel:true, face:'square', fillColor:'yellow', txtokeColor:'black', shadow:true});\r\n\r\n\r\n//Roots\r\nvar root1Point =brd2.create('point',[function(){\r\n\r\n    if(isNaN(zero1.X())){\r\n\r\n    r1 = 20;\r\n    //console.log('IS NAN' , r1);\r\n}\r\n\r\nelse{\r\n    if(o1 == 2 && a1 != 0){\r\n\r\n        r1 = (-b1+Math.pow(b1*b1-4*a1*c1, 0.5))/(2*a1);\r\n        //console.log('CUADRATIC' , r1);\r\n        //console.log('a' , a1);\r\n        //console.log('b' , b1);\r\n        //console.log('c' , c1);\r\n    }\r\n    else if(a1 == 0){\r\n        r1 = -c1/b1;\r\n        //console.log('LINEAL' , r1);\r\n    }\r\n    else{\r\n        r1 = 20;\r\n        //console.log('IDK' , r1);\r\n    }\r\n}\r\n\r\nreturn r1;\r\n\r\n}, 0],{visible: false, size:3, name:'X_1', withLabel:true, fillColor:'yellow', strokeColor:'black', shadow:true, label: {color:'blue', anchorX:'middle', offset:[0, -15], fontSize:function(){return 14*brd2.canvasHeight/800}, cssStyle:'fontFamily:Oswald;'}});\r\n\r\n\r\nvar root2Point =brd2.create('point',[function(){\r\n\r\n    if(isNaN(zero1.X())){\r\n\r\n    r2 = 20;\r\n    //console.log('IS NAN' , r2);\r\n}\r\n\r\nelse{\r\n    if(o1 == 2 && a1 != 0){\r\n\r\n        r2 = (-b1-Math.pow(b1*b1-4*a1*c1, 0.5))/(2*a1);\r\n        //console.log('CUADRATIC' , r2);\r\n        //console.log('a' , a1);\r\n        //console.log('b' , b1);\r\n        //console.log('c' , c1);\r\n    }\r\n    else{\r\n        r2 = 20;\r\n        //console.log('IDK' , r2);\r\n    }\r\n}\r\n\r\nreturn r2;\r\n\r\n}, 0],{visible: false, size:3, name:'X_2', withLabel:true, fillColor:'yellow', strokeColor:'black', shadow:true, label: {anchorX:'middle', offset:[0, -15], color:'blue', fontSize:function(){return 14*brd2.canvasHeight/800}, cssStyle:'fontFamily:Oswald;'}});\r\n\r\n\r\n//Tetxs of polynomials\r\nbrd2.create('text', [-5, 8, 'Your Polynomial'],{display:'internal', highlight:false, fixed: true, anchorX: 'middle', anchorY: 'bottom', CssStyle:'fontFamily:Oswald;fontWeight:bold',fontSize:function(){return Math.round(26*brd2.canvasWidth/800.)}});\r\nvar polynomial11 = brd2.create('text', [-5, 7, function(){return  'f(x) = ' + txta1 + txtb1 + txtc1 }],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(24*brd2.canvasWidth/800.)}},);\r\nvar txta11 ='';\r\nvar txtb11 ='';\r\nvar txtc11 ='';\r\nvar polynomial12 = brd2.create('text', [-5, -1, function(){\r\n\r\n    if(isNaN(zero1.X())){\r\n      return '';\r\n    }\r\n\r\n    else{\r\n\r\n        if(a1== 0){\r\n            txta11 = '';\r\n        }\r\n        else{\r\n            txta11 = a1 + ' (' + root1Point.X().toFixed(2)+ ')^' + o1 + ' ';\r\n        }\r\n\r\n        if(b1== 0){\r\n            txtb11 = '';\r\n        }\r\n        else if(b1 > 0){\r\n            txtb11 = ' + '  + b1 + ' (' + root1Point.X().toFixed(2)+ ') ';\r\n        }\r\n        else{\r\n            txtb11 = b1 + ' (' + root1Point.X().toFixed(2)+ ') ';\r\n        }\r\n\r\n        if(c1== 0){\r\n            txtc11 = '';\r\n\r\n        }\r\n        else if(c1 > 0){\r\n            txtc11 = ' + '  + c1;\r\n\r\n        }\r\n        else{\r\n            txtc11 = c1;\r\n\r\n        }\r\n        return 'f(X_1) = ' + txta11 + txtb11 + txtc11 + ' = 0';\r\n    }\r\n\r\n    }],{visible: false, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(14*brd2.canvasWidth/500.)}},\r\n);\r\n\r\n\r\nvar polynomial13 = brd2.create('text', [-5, -3, function(){\r\n\r\n    if(isNaN(zero1.X())){\r\n      return '';\r\n    }\r\n\r\n    else{\r\n\r\n        if(a1== 0){\r\n            txta11 = '';\r\n        }\r\n        else{\r\n            txta11 = a1 + ' (' + root2Point.X().toFixed(2)+ ')^' + o1 + ' ';\r\n        }\r\n\r\n        if(b1== 0){\r\n            txtb11 = '';\r\n        }\r\n        else if(b1 > 0){\r\n            txtb11 = ' + '  + b1 + ' (' + root2Point.X().toFixed(2)+ ') ';\r\n        }\r\n        else{\r\n            txtb11 = b1 + ' (' + root2Point.X().toFixed(2)+ ') ';\r\n        }\r\n\r\n        if(c1== 0){\r\n            txtc11 = '';\r\n\r\n        }\r\n        else if(c1 > 0){\r\n            txtc11 = ' + '  + c1;\r\n\r\n        }\r\n        else{\r\n            txtc11 = c1;\r\n\r\n        }\r\n        return 'f(X_2 ) = ' + txta11 + txtb11 + txtc11 + ' = 0';\r\n    }\r\n\r\n    }],{visible: false, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(14*brd2.canvasWidth/500.)}},\r\n);\r\n\r\n\r\nvar pDragMe = brd2.create('glider', [-10 , 5, graph],{name:'P(Drag me!)', fillColor:'white', strokeColor:'black', size:3, attractors: [root1Point, root2Point], attractorDistance:0.75, snatchDistance:0.75, withLabel:true, fixed:false, label:{visible:false, color:'blue', offset:[-10,20], cssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(16*brd2.canvasWidth/800.)}}});\r\n//brd2.create('point', [0, ()=>pDragMe.Y()], {name:'y', size:1, color:'black',label:{color:'red', offset:[5,10],cssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(16*brd2.canvasWidth/800.)}}});\r\n//brd2.create('point', [()=>pDragMe.X(), 0], {name:'x', size:1, color:'black',label:{color:'red', offset:[0, -15],cssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(16*brd2.canvasWidth/800.)}}});\r\nbrd2.create('segment', [pDragMe, [0, ()=>pDragMe.Y()]],{dash:1, strokeWidth:1, strokeColor:'grey'});\r\nbrd2.create('segment', [pDragMe, [()=>pDragMe.X(), 0]],{dash:1, strokeWidth:1, strokeColor:'grey'});\r\n//\r\nbrd2.create('text', [-5, 5, 'Value @ Point P'],{display:'internal', highlight:false, fixed: true, anchorX: 'middle', anchorY: 'bottom', CssStyle:'fontFamily:Oswald;fontWeight:bold',fontSize:function(){return Math.round(26*brd2.canvasWidth/800.)}});\r\n\r\n//\r\nvar pDragMetxt = brd2.create('text', [-5, 4, function(){\r\n\r\n    if(isNaN(zero1.X())){\r\n        return '';\r\n      }\r\n\r\n      else{\r\n          return ' y = f(x =' + (pDragMe.X()).toFixed(2) + ') = ' + (a1*Math.pow(pDragMe.X(),o1)+b1*pDragMe.X() + c1).toFixed(2)\r\n        }\r\n}],{fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(24*brd2.canvasWidth/800.)}},);\r\n\r\n\r\n//Tetx of the X roots\r\nvar x1txt = brd2.create('text', [-6.5, 2.5, function(){return 'X_1 = ' + root1Point.X().toFixed(2)}],{fontColor:'blue', visible: false, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(20*brd2.canvasWidth/800.)}},);\r\nvar x2txt = brd2.create('text', [-4, 2.5, function(){return 'X_2 = ' + root2Point.X().toFixed(2)}],{fontColor:'blue', visible: false, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(20*brd2.canvasWidth/800.)}},);\r\n\r\n//Roots appear\r\nsetInterval(checkRoot1, 100);\r\nfunction checkRoot1(){\r\n\r\n    if(pDragMe.X() == root1Point.X() )\r\n    {\r\n        root1Point.setAttribute({visible: true});\r\n        x1txt.setAttribute({visible: true});\r\n\r\n    }\r\n\r\n}\r\n\r\nsetInterval(checkRoot2, 100);\r\nfunction checkRoot2(){\r\n\r\n    if(pDragMe.X() == root2Point.X() )\r\n    {\r\n        root2Point.setAttribute({visible: true});\r\n        x2txt.setAttribute({visible: true});\r\n\r\n    }\r\n\r\n}\r\n/*\r\npDragMe.on('over', checkRoot1);\r\npDragMe.on('over', checkRoot2);\r\n*/\r\n\r\n//REDO AND UNDO\r\nvar redo = brd2.create('image', ['/assets/redo.svg', [2.25, -5], [1.5 , 1.5]], {visible:()=>i<=0, fixed: true});\r\nvar undo = brd2.create('image', ['/assets/undo.svg', [function(){return -redo.X()-1.5}, function(){return redo.Y()}], [1.5 , 1.5]], {visible:()=>i>=1, fixed: true});\r\n\r\npDragMe.on('over', function () {this.label.setAttribute({visible:true});});\r\npDragMe.on('out', function () {this.label.setAttribute({visible:false});});\r\n//\r\nredo.setLabel('Next')\r\nredo.label.setAttribute({visible:false, offset:[15, -15], CssStyle:'fontFamily:Oswald', fontSize:16});\r\nredo.on('over', function () {this.label.setAttribute({visible:true});});\r\nredo.on('out', function () {this.label.setAttribute({visible:false});});\r\n\r\nundo.setLabel('Previous')\r\nundo.label.setAttribute({visible:false, offset:[15, -15], CssStyle:'fontFamily:Oswald', fontSize:16});\r\nundo.on('over', function () {this.label.setAttribute({visible:true});});\r\nundo.on('out', function () {this.label.setAttribute({visible:false});});\r\n\r\n\r\nvar i =0;\r\nvar maxSteps = 1;\r\n\r\nfunction updateStepTexts(){\r\n    eraseText();\r\n\r\n    if (i<=0){\r\n        polynomial11.setAttribute({visible: true});\r\n\t\t  pDragMetxt.setAttribute({visible: true});\r\n\t\t//x1txt.setAttribute({visible:true});\r\n        //x2txt.setAttribute({visible:true});\r\n\r\n    }\r\n    if(i==1){\r\n\r\n        polynomial11.setAttribute({visible: true});\r\n        polynomial12.setAttribute({visible: true});\r\n        polynomial13.setAttribute({visible: true});\r\n        pDragMetxt.setAttribute({visible: true});\r\n\t\t//x1txt.setAttribute({visible:true});\r\n        //x2txt.setAttribute({visible:true});\r\n\r\n    }\r\n}\r\n\r\nfunction eraseText(){\r\n\r\n    polynomial11.setAttribute({visible: false});\r\n    polynomial12.setAttribute({visible: false});\r\n    polynomial13.setAttribute({visible: false});\r\n    pDragMetxt.setAttribute({visible: false});\r\n    pDragMe.moveTo([-10,5], 10);\r\n //   x1txt.setAttribute({visible:false});\r\n //   x2txt.setAttribute({visible:false});\r\n\r\n}\r\n\r\nredo.on('down',function(){\r\n    i = i+1;\r\n\tvis+=1;\r\n    if (i > maxSteps){\r\n        i = maxSteps;\r\n\t\tvis=maxSteps;\r\n    }\r\n    updateStepTexts();\r\n});\r\n\r\nundo.on('down',function(){\r\n    i=i-1;\r\n\tvis-=1;\r\n    if (i < 0){\r\n        i = 0;\r\n\t    vis=0;\r\n    }\r\n    eraseText();\r\n    updateStepTexts();\r\n});\r\n\r\n\r\n\r\n/*\r\nvar root0 = function(){\r\n\r\n    var value=nerdamer(roots[0].totxting()).evaluate().text();\r\n\r\n    if(value.includes('i')){\r\n        return 0;\r\n    }\r\n\r\n    else{\r\n        return value;\r\n    }\r\n}\r\n\r\n\r\nvar root1 = function(){\r\n\r\n    var value=nerdamer(roots[1].totxting()).evaluate().text();\r\n\r\n    if(value.includes('i')){\r\n        return 1;\r\n    }\r\n\r\n    else{\r\n        return value;\r\n    }\r\n}\r\n\r\n\r\nvar zero2 = brd2.create('point', [root0,0],{face:'o' , name: function(){if(isNaN(zero1.Y())){\r\n                                                                            return '';\r\n                                                                        } else{ return 'ROOT';}},  size:4, withLabel:true, face:'square', fillColor:'yellow', txtokeColor:'black', shadow:true});\r\n\r\nvar zero3 = brd2.create('point', [root1,0],{face:'o' , name: 'aa',  size:4, withLabel:true, face:'square', fillColor:'yellow', txtokeColor:'black', shadow:true});\r\n\r\n*/\r\n\r\n\r\nbrd2.unsuspendUpdate();\r\n\r\n}\r\n}\r\nexport default Boxes;\r\n","<template>\r\n  <div>\r\n    <h3 ref=\"intro\">\r\n      Zero of a Polynomial\r\n    </h3>\r\n    <p>\r\n      Zeros of a polynomial, say \\(p(x)\\),  are the values of variable \\(x\\) at which\r\n      the polynomial becomes zero.\r\n      Thus, if \\(x=x_0\\) is a zero of the polynomial \\(p(x)\\) then it must satisfy the following:\r\n      $$p(x=x_0) = 0$$\r\n      The zeros of a polynomial are sometimes also called its roots.\r\n      Depending on the degree of the polynomial, it can have one or more than one zeros &mdash;\r\n      as we will discuss in the following section.\r\n    </p>\r\n    <h3 ref=\"formal\">\n      Number of Zeros of a Polynomial\n    </h3>\r\n    <p>\n      The fundamental theorem of algebra states that a non-zero polynomial has\r\n      the same number of zeros (or roots) as its degree.\r\n      This requires that we account for multiplicity of zeros (i.e. repeated zeros).\r\n      Thus, a polynomial of degree \\(n\\) will have \\(n\\) zeros or roots. Some of these zeros may be repeated or complex.\r\n      Below are few specific examples.\r\n      <ul class=\"a\">\r\n        <li> <h5> Linear Polynomial (Degree =1) </h5> </li>\r\n        A linear polynomial (i.e. a polynomial of degree 1) will have one zero.\r\n        To give an example, let's consider the linear polynomial $$p(x) = 2x-6$$\r\n        This polynomial has exactly one zero, which is \\(x=3\\).\r\n        <li> <h5>Quadratic Polynomial (Degree =2)</h5> </li>\r\n        A quadratic polynomial (i.e. a polynomial of degree 2) will have two zeros\r\n        (note that sometimes these zeros may be repeated).\r\n        Let's first discuss the example of a quadratic polynomial that has two distinct zeros.\r\n        $$q(x) = x^2 -6x +8$$\r\n        After a little rearrangement, \\(q(x)\\) can be rewritten as:\r\n        $$q(x) = (x-4)(x-2)$$\r\n        You can see that this polynomial becomes zero at \\(x=2\\) and at \\(x=4\\). These are the two zeros of this polynomial.\r\n        <br> Next, we discuss the example of a quadratic polynomial that has two repeated (or identical) zeros.\r\n        Consider the following polynomial:\r\n        $$r(x) = x^2 -4x +4$$\r\n        This polynomial can be rearranged as follows:\r\n        $$r(x) = (x-2)(x-2)$$\r\n        Thus, this polynomial has two repeated zeros at \\(x =2\\).\r\n      </ul>\r\n    </p>\r\n    <h3 ref=\"playgraph\">\r\n      MagicGraph | Zeros of a Polynomial\r\n    </h3>\r\n    <p>\n      Through this visually interactive MagicGraph, students will learn about the\r\n      zeros of the polynomial, and how to calculate them using the graphical method.\n    </p>\r\n    <h5> To Get Started: </h5>\r\n    <p>\r\n      You are given a polynomial function named f(x).\r\n      On this MagicGraph, you will also see the graph representing this polynomial function.\r\n      You goal is to identify the zeros (or roots) of this polynomial using the graphical method (described below).<br>\r\n      Point 'P' is a movable point that can glide along the polynomial curve.\r\n      As you drag this point along the polynomial curve, the MagicGraph will\r\n      automatically calculate and display the value of the polynomial at point P.\r\n      Identify the location (or locations) of point P on the graph at which the polynomial becomes zero.\r\n      These are going to be the zeros (or roots) of the polynomial function f(x).\r\n    </p>\r\n    <h5> To Explore:</h5>\r\n    <p>\r\n      After finding the zeros of one polynomial function, tap on the shuffle button to create a new polynomial function\r\n      Each time you tap on the shuffle button,\r\n      the MagicGraph generates a new polynomial for you to practice with.\r\n      Using the graphical method described above, find the zeros of the polynomial &mdash; practice as many times as you want.\n    </p>\r\n    <v-responsive>\r\n      <v-layout justify-center>\r\n        <div id=\"jxgbox1\" class=\"edliy-box-about\" />\r\n      </v-layout>\r\n    </v-responsive>\r\n  </div>\r\n</template>\r\n<script>\r\nimport Boxes from './Boxes.js'\r\nexport default {\r\n  name: 'ForcesStresses',\r\n  created: function () {\r\n    this.$store.commit('navigation/resetState');\r\n    let title = 'Zeros of a Polynomial';\r\n    this.$store.commit('navigation/changeTitle', title);\r\n    this.$store.commit('navigation/changeMenu', title);\r\n    let newTopics = [\r\n      {title: 'zeros of a Polynomial', img:'/assets/number-1.svg', action: () => this.goto('intro')},\r\n      {title: 'Number of zeros', img:'/assets/number-2.svg', action: () => this.goto('formal')},\r\n      {title: 'MagicGraph', img:'/assets/touch.svg', action: () => this.goto('graphical')},\r\n    ];\r\n    this.$store.commit('navigation/replaceTopics', newTopics);\r\n    let newshowhome = false;\r\n    this.$store.commit('navigation/toggleshowhome', newshowhome);\r\n    let newMath =true;\r\n    this.$store.commit('navigation/replaceMath', newMath);\r\n    let newLeftArrow =true;\r\n    this.$store.commit('navigation/replaceLeftArrow', newLeftArrow);\r\n    let newModule=true;\r\n    this.$store.commit('navigation/replaceModule', newModule);\r\n  },\r\n  mounted () {\r\n    MathJax.Hub.Queue([\"Typeset\", MathJax.Hub]);\r\n    Boxes.box1();\r\n  },\r\n  metaInfo() {\r\n  return{ title: 'Roots of a Polynomial',\r\n          titleTemplate: '%s - Learn interactively',\r\n          meta: [ {name: 'viewport', content: 'width=device-width, initial-scale=1'},\r\n                  {name: 'description', content: 'Learn interactively how to draw a free body diagram'}\r\n                ]\r\n        }\r\n   },\r\n}\r\n</script>\r\n\r\n<style lang=\"scss\">\r\n@import 'src/styles/edliy-box.scss';\r\n@import 'src/styles/subtopic-menu.scss';\r\n@import 'src/styles/edliy-box-about.scss';\r\n</style>\r\n","import mod from \"-!../../../../../node_modules/cache-loader/dist/cjs.js??ref--12-0!../../../../../node_modules/thread-loader/dist/cjs.js!../../../../../node_modules/babel-loader/lib/index.js!../../../../../node_modules/cache-loader/dist/cjs.js??ref--0-0!../../../../../node_modules/vue-loader/lib/index.js??vue-loader-options!./Lesson.vue?vue&type=script&lang=js&\"; 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