// Numbas version: finer_feedback_settings
{"name": "J. Richard's copy of Extract common factors of polynomials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Factorise polynomials by identifying common factors. The first&nbsp;expression has a constant common factor; the rest have common factors involving variables.</p>"}, "variables": {"x7": {"definition": "random(1..5)", "description": "", "name": "x7", "group": "Ungrouped variables", "templateType": "anything"}, "x2": {"definition": "random(1..5)", "description": "", "name": "x2", "group": "Ungrouped variables", "templateType": "anything"}, "x1": {"definition": "random(1..5)", "description": "", "name": "x1", "group": "Ungrouped variables", "templateType": "anything"}, "x6": {"definition": "random(1..5)", "description": "", "name": "x6", "group": "Ungrouped variables", "templateType": "anything"}, "a": {"definition": "repeat(random([3, 11, 17, 29, 37, 43] except b),50)", "description": "<p>Vector of every other random prime number</p>", "name": "a", "group": "Ungrouped variables", "templateType": "anything"}, "x4": {"definition": "random(-5..-1)\n\n", "description": "", "name": "x4", "group": "Ungrouped variables", "templateType": "anything"}, "c": {"definition": "repeat(random([ 5,  19, 47] ),50)", "description": "<p>extra primes for when you need a third constant</p>", "name": "c", "group": "Ungrouped variables", "templateType": "anything"}, "x5": {"definition": "random(1..5)", "description": "", "name": "x5", "group": "Ungrouped variables", "templateType": "anything"}, "x3": {"definition": "random(1..5 except [x2])", "description": "", "name": "x3", "group": "Ungrouped variables", "templateType": "anything"}, "b": {"definition": "repeat(random(2, 7, 13, 23, 31, 41, 53),50)", "description": "<p>Vector of the other every other random prime number</p>", "name": "b", "group": "Ungrouped variables", "templateType": "anything"}}, "parts": [{"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "gaps": [{"type": "jme", "vsetRangePoints": 5, "showCorrectAnswer": true, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "marks": 1, "checkVariableNames": false, "variableReplacements": [], "scripts": {"mark": {"script": "question.mark_factorised(this);", "order": "after"}}, "showPreview": true, "vsetRange": [0, 1], "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "answer": "2({a[0]}x+{b[0]})", "showFeedbackIcon": true, "failureRate": 1, "unitTests": [], "expectedVariableNames": []}], "scripts": {}, "sortAnswers": false, "prompt": "<p><em></em>$\\simplify{{2*a[0]}x+{2*b[0]}}=$ [[0]]</p>", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "unitTests": [], "type": "gapfill"}, {"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "gaps": [{"type": "jme", "vsetRangePoints": 5, "showCorrectAnswer": true, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "marks": 1, "checkVariableNames": false, "variableReplacements": [], "scripts": {"mark": {"script": "question.mark_factorised(this);", "order": "after"}}, "showPreview": true, "vsetRange": [0, 1], "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "answer": "6*y*({a[1]}+{b[1]}*y)", "showFeedbackIcon": true, "failureRate": 1, "unitTests": [], "expectedVariableNames": []}], "scripts": {}, "sortAnswers": false, "prompt": "<p><em></em>$\\simplify{6{a[1]}y+6{b[1]}y^2}=$ [[0]]<em></em></p>", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "unitTests": [], "type": "gapfill"}, {"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "gaps": [{"type": "jme", "vsetRangePoints": 5, "showCorrectAnswer": true, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "marks": 1, "checkVariableNames": false, "variableReplacements": [], "scripts": {"mark": {"script": "question.mark_factorised(this);", "order": "after"}}, "showPreview": true, "vsetRange": [0, 1], "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "answer": "x*y*z*({a[2]}+{b[2]}x*y*z)", "showFeedbackIcon": true, "failureRate": 1, "unitTests": [], "expectedVariableNames": []}], "scripts": {}, "sortAnswers": false, "prompt": "<p><em></em>$\\simplify{{a[2]}x*y*z+{b[2]}x^2y^2z^2}=$ [[0]]<em></em></p>", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "unitTests": [], "type": "gapfill"}, {"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "gaps": [{"type": "jme", "vsetRangePoints": 5, "showCorrectAnswer": true, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "marks": 1, "checkVariableNames": false, "variableReplacements": [], "scripts": {"mark": {"script": "question.mark_factorised(this);", "order": "after"}}, "showPreview": true, "vsetRange": [0, 1], "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "answer": "5({a[3]}d+{b[3]}r+m)", "showFeedbackIcon": true, "failureRate": 1, "unitTests": [], "expectedVariableNames": []}], "scripts": {}, "sortAnswers": false, "prompt": "<p><em></em>$\\simplify{5{a[3]}d+5{b[3]}r+5m}=$ [[0]]<em></em></p>", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "unitTests": [], "type": "gapfill"}, {"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "gaps": [{"type": "jme", "vsetRangePoints": 5, "showCorrectAnswer": true, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "marks": 1, "checkVariableNames": false, "variableReplacements": [], "scripts": {"mark": {"script": "question.mark_factorised(this);", "order": "after"}}, "showPreview": true, "vsetRange": [0, 1], "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "answer": "6c*d*({a[4]}d+{b[4]}c+{c[1]}c*d)", "showFeedbackIcon": true, "failureRate": 1, "unitTests": [], "expectedVariableNames": []}], "scripts": {}, "sortAnswers": false, "prompt": "<p><em></em>$\\simplify{6{a[4]}c*d^2+6{b[4]}c^2d+6{c[1]}c^2d^2}=$ [[0]]</p>", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "unitTests": [], "type": "gapfill"}], "variable_groups": [], "rulesets": {}, "variablesTest": {"maxRuns": "1000", "condition": ""}, "statement": "<p>An expression can be factorised by finding common factors of&nbsp;each term in the expression.</p>\n<p>Completely factorise the following expressions by finding their common factors.</p>\n<p><em>Make sure that you include a multiplication symbol <code>*</code> between each algebraic variable, and before brackets, e.g. <code>a*b*(x+1)</code> instead of <code>ab(x+1)</code>. Otherwise, the system might not accept your answer.</em></p>", "tags": ["common factors", "common factors of linear algebraic equations", "common factors of quadratic equations", "finding common factors", "Linear equations", "linear equations", "quadratic equations", "Quadratic Equations", "Quadratic equations", "taxonomy"], "extensions": [], "advice": "<p>In order to factorise the expressions, the factors that make up each term in the expression need to be identified and, where these&nbsp;factors are the same for all terms in the expression, those factors can be taken outside the brackets.&nbsp;Stop when the remaining terms have no more common factors.</p>\n<h4>a)</h4>\n<p>Both terms have a common factor of $2$.</p>\n<p>\\begin{align}<br/>\\simplify{2{a[0]}x+2{b[0]}}&amp;=<br/>(\\simplify[]{2{a[0]}})x+2\\times\\var{b[0]}\\\\<br/>&amp;=\\simplify[]{2({a[0]}x+{b[0]})}<br/>\\end{align}</p>\n<h4>b)</h4>\n<p>Both terms have common factors of $6$ and $y$.</p>\n<p>\\begin{align}<br/>\\simplify{6{a[1]}y+6{b[1]}y^2}&amp;= 6 \\times \\var{a[1]} y + 6 \\times \\var{b[1]} y^2 \\\\<br/>&amp;= 6 \\times (\\simplify{{a[1]}y + {b[1]}y^2}) \\\\<br/>&amp;=6y(\\simplify[]{{a[1]}+{b[1]}y})<br/>\\end{align}</p>\n<h4>c)</h4>\n<p>Both terms have&nbsp;common factors of $x$, $y$ and $z$.</p>\n<p>\\begin{align}<br/>\\simplify{{a[2]}x*y*z+{b[2]}x^2y^2z^2}&amp;=\\var{a[2]} \\times xyz + \\var{b[2]} \\times xyz \\times xyz\\\\<br/>&amp;=xyz(\\var{a[2]} + \\var{b[2]} xyz)<br/>\\end{align}</p>\n<h4>d)</h4>\n<p>All three terms have a common factor of $5$.</p>\n<p>\\begin{align}<br/>\\simplify{5{a[3]}d+5{b[3]}r+5m}&amp;= 5 \\times \\var{a[3]} d+5 \\times \\var{b[3]} r + 5 m \\\\<br/>&amp;=\\simplify[]{5({a[3]}d+{b[3]}r+m)}<br/>\\end{align}</p>\n<h4>e)</h4>\n<p>All the terms have common factors of $6$, $c$ and $d$.</p>\n<p><em></em>\\begin{align}<br/>\\simplify{6{a[4]}cd^2+6{b[4]}c^2d+6{c[1]}c^2d^2} &amp;= 6 \\times \\var{a[4]} c d^2 \\;+\\; 6 \\times \\var{b[4]} c^2 d \\;+\\; 6 \\times \\var{c[1]} c^2 d^2 \\\\<br/>&amp;= 6(\\var{a[4]} c d^2 + \\var{b[4]} c^2 d + \\var{c[1]} c^2 d^2) \\\\<br/>&amp;=6cd(\\var{a[4]}d+\\var{b[4]}c+\\var{c[1]}cd)<br/>\\end{align}</p>\n<h4></h4>", "name": "J. Richard's copy of Extract common factors of polynomials", "functions": {}, "ungrouped_variables": ["a", "b", "c", "x1", "x2", "x3", "x4", "x5", "x6", "x7"], "preamble": {"js": "question.mark_factorised = function(part) {\n  var match = Numbas.jme.display.matchExpression;\n  var unwrap = Numbas.jme.unwrapValue;\n  var getCommutingTerms = Numbas.jme.display.getCommutingTerms;\n  var matchTree = Numbas.jme.display.matchTree;\n\n  var expr = part.studentAnswer;\n\n  // is the student's answer in the form `factors*(terms)`?\n  var m = match('m_all(m_any(m_number,m_type(name),m_type(name)^m_number))*(m_all(??)+m_nothing);rest', expr, true);\n  if(!m) {\n    part.multCredit(0,\"You don't seem to have extracted a common factor.\");\n    return;\n  }\n  \n  var terms = getCommutingTerms(m.rest,'+').terms;\n  \n  var gcd;  // greatest common divisor of the coefficients of each term\n  var min_degrees = {}; // minimum degree of each variable across the terms - if greater than 0, that variable is a common factor\n  \n  // for each term, collect the scalar part and the degree of each of the variables\n  terms.map(function(t,i) { \n    var factors = getCommutingTerms(t,'*').terms;\n    var degrees = {};\n    var n = 1;\n    factors.forEach(function(f) {\n      var tok = f.tok;\n      var scalar = 1;\n      var name = null;\n      var degree = 0;\n      var m = Numbas.jme.display.matchTree(Numbas.jme.compile('m_any(m_number;n, m_type(name);name, m_number;n*m_type(name);name, m_type(name);name^(m_number;degree), m_number;n*m_type(name);name^m_number;degree )'),f,true)\n\n      console.log(Numbas.jme.display.treeToJME(f));\n\n      if(!m) {\n        console.log('  no match');\n        return;\n      }\n      console.log('  match',m.n,m.name,m.degree);\n      if(m.n) {\n        n *= unwrap(m.n.tok);\n      }\n      if(m.name) {\n        name = m.name.tok.name;\n        if(m.degree) {\n          degree = unwrap(m.degree.tok);\n        } else {\n          degree = 1;\n        }\n        degrees[name] = (degrees[name] || 0) + degree;\n      }\n    });\n    console.log(Numbas.jme.display.treeToJME(t), degrees, n);\n      \n    if(i==0) {\n      gcd = n;\n    } else {\n      gcd = Numbas.math.gcd(gcd,n);\n    }\n    \n    Object.keys(degrees).forEach(function(k) {\n      if(i==0) {\n        min_degrees[k] = degrees[k];\n      } else {\n        min_degrees[k] = Math.min(min_degrees[k] || 0, degrees[k]);\n      }\n    });\n    Object.keys(min_degrees).forEach(function(k) {\n      if(degrees[k]===undefined) {\n        min_degrees[k] = 0;\n      }\n    });\n    \n  });\n  \n  var common_factors = [];\n  if(gcd>1) {\n    common_factors.push(gcd);\n  }\n  Object.keys(min_degrees).filter(function(k){return min_degrees[k]>0}).forEach(function(k) {\n    common_factors.push(k+'^'+min_degrees[k])\n  });\n  \n  if(common_factors.length==0) {\n    return true;\n  } else {\n    part.multCredit(0,\"Your terms still have a common factor of $\"+Numbas.jme.display.exprToLaTeX(common_factors.join('*'),'all',Numbas.jme.builtinScope)+\"$.\");\n  }\n};", "css": ""}, "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}, {"name": "J. Richard Snape", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1700/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}, {"name": "J. Richard Snape", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1700/"}]}