// Numbas version: exam_results_page_options
{"name": "Polynomials: Expand and Simplify 1d", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Polynomials: Expand and Simplify 1d", "tags": [], "metadata": {"description": "<p>Calculate the product of two quadratic expressions of the form $ax^2+bx+c$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Given two functions, \\[ f(x) = \\simplify{{a1}x^2+{b1}x+{c1}}, \\quad g(x)=\\simplify{{a2}x^2+{b2}x+{c2}},\\]</p>\n<p>calculate \\[ f(x)g(x).\\]</p>", "advice": "<p>To split this calculation into stages, we will first multiply $g(x)$ by each term in $f(x)$:</p>\n<p>\\[ \\begin{split}&nbsp;f(x)g(x)&nbsp;&amp;\\,= (\\simplify{{a1}x^2+{b1}x+{c1}})(\\simplify{{a2}x^2+{b2}x+{c2}}) \\\\ &amp;\\,= \\simplify{{a1}x^2 ({a2}x^2+{b2}x+{c2}) +{b1}x({a2}x^2+{b2}x+{c2})+{c1}({a2}x^2+{b2}x+{c2})} \\end{split} \\]</p>\n<p>We can then expand each of the brackets and collect similar terms:</p>\n<p>\\[ \\begin{split}&nbsp;f(x)g(x) &amp;\\,= \\simplify{{a1}x^2 ({a2}x^2+{b2}x+{c2}) +{b1}x({a2}x^2+{b2}x+{c2})+{c1}({a2}x^2+{b2}x+{c2})} \\\\ &amp;\\,= \\simplify[all,!cancelTerms, !collectNumbers]{{a1*a2}x^4+{a1*b2}x^3+{a1*c2}x^2+{b1*a2}x^3+{b1*b2}x^2+{b1*c2}x+{c1*a2}x^2+{c1*b2}x+{c1*c2}}\\\\ &amp;\\,= \\simplify[all]{{a1*a2}x^4+{a1*b2+b1*a2}x^3+{a1*c2+b1*b2+c1*a2}x^2+{b1*c2+c1*b2}x+{c1*c2}}\\end{split} \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(-3..3 except [0,m,-m])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "a2", "b1", "b2", "c1", "c2", "m", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a1*a2}x^4+{a1*b2+a2*b1}x^3+{a1*c2+a2*c1+b1*b2}x^2+{b1*c2+b2*c1}x+{c1*c2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "`+-$n`?*x^4+`+-$n`?*x^3+`+-$n`?x^2+`+-$n`?*x+`+-$n`?", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}