Practice dividing polynomials using the long division method.
"}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "Group", "pickQuestions": 1, "questions": [{"name": "Q9 - Coordinate Geometry, Line and Parabola", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "David Rickard", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/451/"}], "functions": {}, "ungrouped_variables": ["a", "c", "values", "v1", "m", "yc", "xval"], "tags": ["graph", "interactive", "Jsxgraph", "jsxgraph", "plot", "quadratic"], "advice": "This is the graph you should have obtained.
\n", "rulesets": {"std": ["all", "fractionNumbers"]}, "parts": [{"prompt": "What is the slope and the y-axis intercept of the line $\\simplify{y={m}x+{yc}}$ ?
\nSlope = [[0]] y-intercept = [[1]]
\nWhat will the $y$-coordinate of the point on the line whose $x$-coordinate is $\\var{xval}$?
\n( $\\var{xval}$ , [[2]] )
\n\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "m", "minValue": "m", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": "0.5", "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": true, "variableReplacements": [], "maxValue": "yc", "minValue": "yc", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": "0.5", "type": "numberentry", "showPrecisionHint": false}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": false, "scripts": {}, "answer": "{xval}*{m}+{yc}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "Fill in the table of values for $y=\\simplify[std]{{a}x^2+{c}}$:
\n| $x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|---|---|---|
| $y$ | \n[[0]] | \n[[1]] | \n[[2]] | \n[[3]] | \n[[4]] | \n[[5]] | \n[[6]] | \n
Slide the points to the correct $y$ values.
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{values[0]}", "minValue": "{values[0]}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": "0.5", "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{values[1]}", "minValue": "{values[1]}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": "0.5", "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{values[2]}", "minValue": "{values[2]}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": "0.5", "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{values[3]}", "minValue": "{values[3]}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": "0.5", "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{values[4]}", "minValue": "{values[4]}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": "0.5", "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{values[5]}", "minValue": "{values[5]}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": "0.5", "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{values[6]}", "minValue": "{values[6]}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": "0.5", "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "You are given the quadratic formula
\n$y=\\simplify[std]{{a}x^2+{c}}$
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "table#values th {\n background: none;\n text-align: center;\n}", "js": "\nfunction dragpoint_board() {\n\n var scope = question.scope; \n var a = scope.variables.a.value;\n\n var c = scope.variables.c.value;\n var maxy = Math.max(Math.abs(a*9+c),Math.abs(c));\n \n var div = Numbas.extensions.jsxgraph.makeBoard('250px','400px',{boundingBox:[-5,maxy+3,5,-maxy-3],grid:true});\n $(question.display.html).find('#dragpoint').append(div);\n \n var board = div.board;\n \n //shorthand to evaluate a mathematical expression to a number\n function evaluate(expression) {\n try {\n var val = Numbas.jme.evaluate(expression,question.scope);\n return Numbas.jme.unwrapValue(val);\n }\n catch(e) {\n // if there's an error, return no number\n return NaN;\n }\n }\n \n // set up points array\n var num_points = 7;\n var points = [];\n \n \n // this function sets up the i^th point\n function make_point(i) {\n \n // calculate initial coordinates\n var x = i-(num_points-1)/2;\n \n // create an invisible vertical line for the point to slide along\n var line = board.create('line',[[x,0],[x,1]],{visible: false});\n \n // create the point\n var point = points[i] = board.create(\n 'glider',\n [i-(num_points-1)/2,0,line],\n {\n name:'',\n size:2,\n snapSizeY: 0.1, // the point will snap to y-coordinates which are multiples of 0.1\n snapToGrid: true\n }\n );\n \n // the contents of the input box for this point\n var studentAnswer = question.parts[2].gaps[i].display.studentAnswer;\n \n //Here I have commented out the functions which connect the student input to the graph and the filling in of the answer fields\n //when the student drags the points on the graph.\n \n // watch the student's input and reposition the point when it changes. \n // ko.computed(function() {\n // y = evaluate(studentAnswer());\n //if(!(isNaN(y)) && board.mode!=board.BOARD_MODE_DRAG) {\n // point.moveTo([x,y],100);\n // }\n // });\n \n // when the student drags the point, update the gapfill input\n point.on('drag',function(){\n var y = Numbas.math.niceNumber(point.Y());\n studentAnswer(y);\n });\n \n }\n \n // create each point\n for(var i=0;iDisconnected the graph from the answer fields.
", "description": "Compute a table of values for a quadratic function. The student input is now disconnected from the graph so that they slide the points on the graph after they input the values and the answer fields are not updated. Now includes a graph in advice.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Graphing: nth degree polynomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "functions": {}, "ungrouped_variables": ["n", "c", "parity", "poly", "case", "lbeh", "rbeh", "poly0"], "tags": ["graphing", "polynomial", "polynomials", "sketching"], "advice": "$\\phantom{a}$
", "rulesets": {}, "parts": [{"stepsPenalty": "1", "prompt": "This equation can be described as a
", "matrix": [0, "0", "0", 0, 0, "1"], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "choices": ["$0$th degree polynomial
", "polynomial of degree $1$
", "polynomial of degree $2$
", "hyperbola
", "circle
", "polynomial of degree $\\var{n}$
"], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "An equation of the form $y=c_nx^n+c_{n-1}x^{n-1}+\\ldots+c_1x+c_0$ is called an $n$th degree polynomial.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "maxMarks": 0, "scripts": {}, "distractors": ["", "", "", "", "", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"stepsPenalty": "1", "prompt": "As we move to the far left of the graph, the graph
", "matrix": "lbeh", "shuffleChoices": false, "variableReplacements": [], "choices": ["goes upwards.
", "goes downwards.
"], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "What happens to the graph as you go far to the left or right is called the long term behaviour of a graph.
\nThe leading term (the term that includes the highest power) determines the long term behaviour of a polynomial. For our polynomial this is $\\simplify{{c[n]}x^{n}}$.
\nAs we go far to the left of the graph $x$ is negative, and so $\\simplify{{c[n]}x^{n}}$ is negative. That is, the graph goes downwards. is positive. That is, the graph goes upwards.
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "maxMarks": 0, "marks": 0, "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"stepsPenalty": "1", "prompt": "As we move to the far right of the graph, the graph
", "matrix": "rbeh", "shuffleChoices": false, "variableReplacements": [], "choices": ["goes upwards.
", "goes downwards.
"], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "What happens to the graph as you go far to the left or right is called the long term behaviour of a graph.
\nThe leading term (the term that includes the highest power) determines the long term behaviour of a polynomial. For our polynomial this is $\\simplify{{c[n]}x^{n}}$.
\nAs we go far to the right of the graph $x$ is negative, and so $\\simplify{{c[n]}x^{n}}$ is negative. That is, the graph goes downwards. is positive. That is, the graph goes upwards.
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "maxMarks": 0, "marks": 0, "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"stepsPenalty": "1", "prompt": "The $y$-intercept of the graph is $y=$[[0]].
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "{c[0]}", "minValue": "{c[0]}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "The $y$-intercept is the value of $y$ when $x=0$, that is, the value of $y$ where the graph hits the $y$-axis. To find it, substitute $x=0$ into our equation {poly}. Doing so shows that $y=\\var{c[0]}$ is the $y$-intercept.
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": "1", "prompt": "Given any polynomial of degree $\\var{n}$, the maximum number of $x$-intercepts in its graph is [[0]] .
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{n}", "minValue": "{n}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "The $x$-intercept is the value of $x$ when $y=0$, that is, the value of $x$ where the graph hits the $x$-axis. To find it, substitute $y=0$ into our equation:
\n{poly0}
\nThe Fundamental Theorem of Algebra says there are exactly $\\var{n}$ (complex) solutions to this equation (including multiplicity). The $x$-intercepts for our polynomial are real solutions to the above equation and therefore there are at most $\\var{n}$ $x$-intercepts.
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": "1", "prompt": "Given any polynomial of degree $\\var{n}$, the maximum number of possible 'bends' or 'turns' in the graph is [[0]].
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{n-1}", "minValue": "{n-1}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "A degree $n$ polynomial has at most $n-1$ bends in its graph.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "You are given the equation {poly}.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"case": {"definition": "if(c[n]>0 and parity=1,1,if(c[n]>0 and parity=0,2,if(c[n]<0 and parity=1,3,4)))", "templateType": "anything", "group": "Ungrouped variables", "name": "case", "description": ""}, "parity": {"definition": "mod(n,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "parity", "description": ""}, "c": {"definition": "repeat(random(0,random(-12..12 except 0)),n)+[random(-12..12 except 0)]", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "rbeh": {"definition": "[if(case=1 or case=2,1,0),if(case=1 or case=2,0,1)]", "templateType": "anything", "group": "Ungrouped variables", "name": "rbeh", "description": ""}, "lbeh": {"definition": "[if(case=3 or case=2,1,0),if(case=3 or case=2,0,1)]", "templateType": "anything", "group": "Ungrouped variables", "name": "lbeh", "description": ""}, "poly0": {"definition": "if(n=2,'\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2}\\$',\nif(n=3, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3}\\$',\nif(n=4, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4}\\$',\nif(n=5, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5}\\$',\nif(n=6, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6}\\$',\nif(n=7, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7}\\$',\nif(n=8, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8}\\$',\nif(n=9, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8+{c[9]}x^9}\\$',\n'\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8+{c[9]}x^9+{c[10]}x^{10}}\\$')))))))) ", "templateType": "anything", "group": "Ungrouped variables", "name": "poly0", "description": ""}, "poly": {"definition": "if(n=2,'\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2}\\$',\nif(n=3, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3}\\$',\nif(n=4, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4}\\$',\nif(n=5, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5}\\$',\nif(n=6, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6}\\$',\nif(n=7, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7}\\$',\nif(n=8, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8}\\$',\nif(n=9, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8+{c[9]}x^9}\\$',\n'\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8+{c[9]}x^9+{c[10]}x^{10}}\\$')))))))) ", "templateType": "anything", "group": "Ungrouped variables", "name": "poly", "description": ""}, "n": {"definition": "random(5..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}}, "metadata": {"description": "Understanding the general facts about polynomials of degree n.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Denis's copy of Factorise a quadratic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Denis Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1216/"}], "variables": {"n5": {"definition": "a*b", "name": "n5"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "f": {"definition": "a*b", "name": "f"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "n3": {"definition": "2*a*b", "name": "n3"}, "disc": {"definition": "(b*c+a*d)^2-4*a*b*c*d", "name": "disc"}, "c": {"definition": "c1*s3", "name": "c"}, "a": {"definition": "random(2..5)", "name": "a"}, "rep": {"definition": "switch(disc=0,'repeated', ' ')", "name": "rep"}, "c1": {"definition": "switch(f=1, random(1..6),f=2,random(1,3,5,7,9),f=3,random(1,2,5,7,8),f=4,random(1,3,5,7,9),f=6, random(1,5,7,8),f=9,random(1,2,4,7,8),f=8,random(1,3,5,7,9),f=12,random(1,5,7),random(1,3,5,7))", "name": "c1"}, "rdis": {"definition": "switch(disc=0,'The discriminant is '+ 0+' and so we get two repeated roots in this case.',disc<0, 'There are no real roots.','The roots exist and are distinct. ')", "name": "rdis"}, "n1": {"definition": "b*c+a*d", "name": "n1"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "d": {"definition": "if(d1=-b*c/a, max(d1+1,random(1..5))*s3,d1*s3)", "name": "d"}, "n2": {"definition": "b*c-a*d", "name": "n2"}, "d1": {"definition": "switch(f=1, random(1..6),f=2,random(1,3,5,7,9),f=3,random(1,2,5,7,8),f=4,random(1,3,5,7,9),f=6, random(1,5,7,8),f=9,random(1,2,4,7,8),f=8,random(1,3,5,7,9),f=12,random(1,5,7),random(1,3,5,7))", "name": "d1"}, "b": {"definition": "random(1..4)", "name": "b"}, "n4": {"definition": "abs(n2)", "name": "n4"}}, "progress": "ready", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "notes": "\n \t\t5/08/2012:
\n \t\tAdded more tags.
\n \t\tAdded description.
\n \t\tAllowed the use of decimals.
\n \t\tImproved display of Advice.
\n \t\t", "description": "Factorise $\\displaystyle{ax ^ 2 + bx + c}$ into linear factors.
"}, "functions": {}, "variable_groups": [], "tags": ["Steps", "factorisation", "factorise a quadratic", "factorization", "factorize a quadratic", "linear factors", "quadratics", "steps"], "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "pickQuestions": 0, "questions": []}], "parts": [{"marks": 0.0, "gaps": [{"notallowed": {"partialcredit": 0.0, "message": "Factorise the expression into two factors.
", "showstrings": false, "strings": ["^", "x*x", "x x", "x(", "x (", ")x", ") x"]}, "marks": 2.0, "answersimplification": "std", "vsetrange": [0.0, 1.0], "answer": "((({a} * x) + {( - c)}) * (({b} * x) + {( - d)}))", "musthave": {"partialcredit": 0.0, "message": "factorise the expression into two factors
", "showstrings": false, "strings": ["(", ")"]}, "checkingaccuracy": 0.0001, "type": "jme", "checkingtype": "absdiff", "vsetrangepoints": 5.0}], "steps": [{"marks": 0.0, "type": "information", "prompt": "\nFactorisation by finding the roots
\nIf you cannot spot a direct factorisation of a quadratic $q(x)$ then finding the roots of the equation $q(x)=0$ can help you.
\nFor if $x=r$ and $x=s$ and are the roots then $q(x)=a(x-r)(x-s)$ for some constant $a$.
\nFinding the roots of a quadratic using the standard formula
We can use the following formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$
The two roots are
\n\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]
there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$
1. $\\Delta \\gt 0$. The roots are real and distinct
\n2. $\\Delta=0$. The roots are real and equal. Their value is $\\frac{-b}{2a}$
\n3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.
\n "}], "stepspenalty": 1.0, "type": "gapfill", "prompt": "\n\\[q(x)=\\simplify[std]{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}\\]
$q(x)=\\;$ [[0]]
You can get more information on factorising a quadratic by clicking on Show steps. You will lose 1 mark if you do so.
\n "}], "statement": "
Factorise the following quadratic expression $q(x)$ into linear factors i.e. input $q(x)$ in the form
\\[(ax+b)(cx+d)\\] for suitable integers $a$, $b$, $c$ and $d$ .
Direct Factorisation.
\nIf you can spot a direct factorisation then this is the quickest way to do this question.
\nFor this example we have the factorisation
\n\\[\\simplify{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d} = ({a} * x + { -c}) * ({b} * x + { -d})}\\]
\nFactorisation by finding the roots.
\nFor if $x=r$ and $x=s$ and are the roots then $q(x)=a(x-r)(x-s)$ where $a$ is the coefficient of $x^2$.
\nThere are several methods of finding the roots – here are the main methods.
\nFinding the roots of a quadratic using the standard formula.
\nWe can use the following formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$
\nThe two roots are
\n\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]
there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$
1. $\\Delta \\gt 0$. The roots are real and distinct
\n2. $\\Delta=0$. The roots are real and equal. Their common value is $-\\frac{b}{2a}$
\n3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.
\nFor this question the discriminant of $\\simplify{{a*b}x^2+{-b*c-a*d}x+{c*d}}$ is $\\Delta = \\simplify{{-(b*c+a*d)}^2-4*{a*b}*{c*d}={disc}}$
\n{rdis}.
\nSo the {rep} roots are:
\n\\[\\begin{eqnarray} x = \\frac{\\var{n1} + \\sqrt{\\var{disc}}}{\\var{n3}} &=& \\frac{\\var{n1} + \\var{n4} }{\\var{n3}} &=& \\simplify{{n1 + n4}/ {n3}}\\\\ x = \\frac{\\var{n1} - \\sqrt{\\var{disc}}}{\\var{n3}} &=& \\frac{(\\var{n1} - \\var{n4}) }{\\var{n3}} &=& \\simplify{{n1 - n4}/ {n3}} \\end{eqnarray}\\]
So we see that:
\\[q(x)=\\simplify{{a*b}}\\left(\\simplify{x-{n1 + n4}/ {n3}}\\right)\\left(\\simplify{x-{n1 - n4}/ {n3}}\\right)=\\simplify{({b} * x + { -d}) * ({a} * x + { -c})}\\]
Completing the square.
\nFirst we complete the square for the quadratic expression $\\simplify{{a*b}x^2+{-n1}x+{c*d}}$
\\[\\begin{eqnarray} \\simplify{{a*b}x^2+{-n1}x+{c*d}}&=&\\var{n5}\\left(\\simplify{x^2+({-n1}/{a*b})x+ {c*d}/{a*b}}\\right)\\\\ &=&\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2+ \\simplify{{c*d}/{a*b}-({-n1}/({2*a*b}))^2}\\right)\\\\ &=&\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2 -\\simplify{ {n2^2}/{4*(a*b)^2}}\\right) \\end{eqnarray} \\]
So to solve $\\simplify{{a*b}x^2+{-n1}x+{c*d}}=0$ we have to solve:
\\[\\begin{eqnarray} \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&\\phantom{{}}& -\\simplify{ {n2^2}/{4*(a*b)^2}}=0\\Rightarrow\\\\ \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&\\phantom{{}}&=\\simplify{ {n2^2}/{4*(a*b)^2}=({abs(n2)}/{2*a*b})^2} \\end{eqnarray}\\]
So we get the two {rep} solutions:
\\[\\begin{eqnarray} \\simplify{x+({-n1}/{2*a*b})}&=&\\simplify{{abs(n2)}/{2*a*b}} \\Rightarrow &x& = \\simplify{({abs(n2)+n1}/{2*a*b})}\\\\ \\simplify{x+({-n1}/{2*a*b})}&=&\\simplify{-({abs(n2)}/{2*a*b})} \\Rightarrow &x& = \\simplify{({n1-abs(n2)}/{2*a*b})} \\end{eqnarray}\\]
Finding these roots then gives the factorisation as before.
Solve for $x$: \\[\\simplify[std]{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}=0\\]
The least root is $x=\\;$ [[0]]. The greatest root is $x=\\;$ [[1]]
You can get more information on solving a quadratic by clicking on Show steps. You will lose 1 mark if you do so.
\nEnter the least root first. If the roots are equal, enter the root in both input boxes.
\nEnter the roots as fractions or integers, not as decimals.
\n ", "steps": [{"marks": 0.0, "prompt": "\nFinding the roots by factorisation.
\nFinding a factorisation of a quadratic $q(x)=a(x-r)(x-s)$ where $a$ is the coefficient of $x^2$ gives the roots $x=r$, $x=s$ immendiately.
\nIf you cannot find a factorisation then there are several other methods you can use.
\nUsing the formula for the roots.
\nYou can find the roots by using the formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$
\nThe two roots are:
\n\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]
there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$
1. $\\Delta \\gt 0$. The roots are real and distinct
\n2. $\\Delta=0$. The roots are real and equal. Their value is $\\displaystyle \\frac{-b}{2a}$
\n3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.
\n\n ", "type": "information"}], "type": "gapfill", "gaps": [{"marks": 1.0, "vsetrangepoints": 5.0, "type": "jme", "notallowed": {"showstrings": false, "message": "
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", "advice": "\nDirect Factorisation
\nIf you can spot a direct factorisation then this is the quickest way to do this question.
\nFor this example we have the factorisation
\n\\[\\simplify{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d} = ({a} * x + { -c}) * ({b} * x + { -d})}\\]
\nHence we find the roots:
\\[\\begin{eqnarray} x&=& \\simplify{{n1-n4}/{2*a*b}}\\\\ x&=& \\simplify{{n1+n4}/{2*a*b}} \\end{eqnarray} \\]
Other Methods.
\nThere are several methods of finding the roots – here are the main methods.
\nFinding the roots of a quadratic using the standard formula.
\nWe can use the following formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$
\nThe two roots are
\n\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]
there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$
1. $\\Delta \\gt 0$. The roots are real and distinct
\n2. $\\Delta=0$. The roots are real and equal. Their common value is $\\displaystyle -\\frac{b}{2a}$
\n3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.
\nFor this question the discriminant of $\\simplify{{a*b}x^2+{-b*c-a*d}x+{c*d}}$ is $\\Delta = \\simplify[std]{{-n1}^2-4*{a*b*c*d}}=\\var{disc}$
\n{rdis}.
\nSo the {rep} roots are:
\n\\[\\begin{eqnarray} x = \\frac{\\var{n1} - \\sqrt{\\var{disc}}}{\\var{n3}} &=& \\frac{\\var{n1} - \\var{n4} }{\\var{n3}} &=& \\simplify{{n1 - n4}/ {n3}}\\\\ x = \\frac{\\var{n1} + \\sqrt{\\var{disc}}}{\\var{n3}} &=& \\frac{\\var{n1} + \\var{n4} }{\\var{n3}} &=& \\simplify{{n1 + n4}/ {n3}} \\end{eqnarray}\\]
\nCompleting the square.
\nFirst we complete the square for the quadratic expression $\\simplify{{a*b}x^2+{-n1}x+{c*d}}$
\\[\\begin{eqnarray} \\simplify{{a*b}x^2+{-n1}x+{c*d}}&=&\\var{n5}\\left(\\simplify{x^2+({-n1}/{a*b})x+ {c*d}/{a*b}}\\right)\\\\ &=&\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2+ \\simplify{{c*d}/{a*b}-({-n1}/({2*a*b}))^2}\\right)\\\\ &=&\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2 -\\simplify{ {n2^2}/{4*(a*b)^2}}\\right) \\end{eqnarray} \\]
So to solve $\\simplify{{a*b}x^2+{-n1}x+{c*d}}=0$ we have to solve:
\\[\\begin{eqnarray} \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&\\phantom{{}}& -\\simplify{ {n2^2}/{4*(a*b)^2}}=0\\Rightarrow\\\\ \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&\\phantom{{}}&=\\simplify{ {n2^2}/{4*(a*b)^2}=({abs(n2)}/{2*a*b})^2} \\end{eqnarray}\\]
So we get the two {rep} solutions:
\\[\\begin{eqnarray} \\simplify{x+({-n1}/{2*a*b})}&=&\\simplify{-{abs(n2)}/{2*a*b}} \\Rightarrow &x& = \\simplify{({-abs(n2)+n1}/{2*a*b})}\\\\ \\simplify{x+({-n1}/{2*a*b})}&=&\\simplify{({abs(n2)}/{2*a*b})} \\Rightarrow &x& = \\simplify{({n1+abs(n2)}/{2*a*b})} \\end{eqnarray}\\]
Solve for $x$: $\\displaystyle ax ^ 2 + bx + c=0$.
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| $\\simplify{x+{a}}$ | \n$\\simplify{{b}*x^2}$ | \n${\\var{xcoeff}}$ | \n$\\var{con}$ | \n
| \n | [[2]] | \n[[3]] | \n\n |
| \n | \n | [[4]] | \n[[5]] | \n
| \n | \n | [[6]] | \n[[7]] | \n
| \n | \n | \n | [[8]] | \n
Write the final answer in the following form:
\n| $\\simplify{{b}*x^2+{c+a*b}*x+{a*c+d}}$ | \n$=$ | \n[[0]] | $+$ | \n[[1]] | \n
| $\\simplify{x+{a}}$ | \n\n | \n | $\\simplify{x+{a}}$ | \n
Divide the polynomial $\\simplify{{b}*x^2+{c+a*b}*x+{a*c+d}}$ by $\\simplify{x+{a}}$.
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| $\\simplify{x+{a}}$ | \n$\\simplify{{b}*x^3}$ | \n${\\var{x2coeff}}$ | \n${\\var{xcoeff}}$ | \n$\\var{con}$ | \n
| \n | [[3]] | \n[[4]] | \n\n | \n |
| \n | \n | [[5]] | \n[[6]] | \n[[7]] | \n
| \n | \n | [[8]] | \n[[9]] | \n\n |
| \n | \n | \n | [[10]] | \n[[11]] | \n
| \n | \n | \n | [[12]] | \n[[13]] | \n
| \n | \n | \n | \n | [[14]] | \n
Write the final answer in the following form:
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| $\\simplify{x+{a}}$ | \n\n | \n | \n | $\\simplify{x+{a}}$ | \n
Divide the polynomial $\\simplify{{b}*x^3+{c+a*b}*x^2+{a*c+d}*x+{a*d+r}}$ by $\\simplify{x+{a}}$.
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| $\\simplify{{a1}*x^2+{a2}*x+{a3}}$ | \n$\\simplify{{a1*b1}*x^3}$ | \n${\\var{x2coeff}}$ | \n${\\var{xcoeff}}$ | \n$\\var{con}$ | \n|
| \n | [[2]] | \n[[3]] | \n[[4]] | \n\n | \n |
| \n | \n | [[5]] | \n[[6]] | \n[[7]] | \n|
| \n | \n | [[8]] | \n[[9]] | \n[[10]] | \n|
| \n | \n | \n | [[11]] | \n[[12]] | \n
Write the final answer in the following form:
\n| $\\simplify{{a1*b1}*x^3+{a1*b2+a2*b1}*x^2+{a2*b2+a3*b1+r1}*x+{a3*b2+r2}}$ | \n$=$ | \n[[0]] | \n$+$ | \n[[1]] | \n
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Divide the polynomial $\\simplify{{a1*b1}*x^3+{a1*b2+a2*b1}*x^2+{a2*b2+a3*b1+r1}*x+{a3*b2+r2}}$ by $\\simplify{{a1}*x^2+{a2}*x+{a3}}$.
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