// Numbas version: finer_feedback_settings
{"name": "Polynomials: Expand and Simplify 2d", "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 2d", "tags": [], "metadata": {"description": "<p>Finding the product of a linear function of the form $mx+c$ and a cubic function of the form $ax^3+bx^2+cx+d$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Expand and simplify:</p>\n<p>\\[ (\\simplify{{c2}x+{d2}})(\\simplify{{a1}x^3+{b1}x^2+{c1}x+{d1}})\\]</p>", "advice": "<p>To expand and simplify&nbsp;</p>\n<p>\\[ (\\simplify{{c2}x+{d2}})(\\simplify{{a1}x^3+{b1}x^2+{c1}x+{d1}}),\\]</p>\n<p>we will first multiply the cubic polynomial by each term in the first bracket:</p>\n<p>\\[ (\\simplify{{c2}x+{d2}})(\\simplify{{a1}x^3+{b1}x^2+{c1}x+{d1}}) = \\simplify{{c2}x ({a1}x^3+{b1}x^2+{c1}x+{d1})+{d2}({a1}x^3+{b1}x^2+{c1}x+{d1})}.\\]</p>\n<p>We can now expand each bracket and collect similar terms to simplify:</p>\n<p>\\[ \\begin{split} (\\simplify{{c2}x+{d2}})(\\simplify{{a1}x^3+{b1}x^2+{c1}x+{d1}}) &amp;\\,= \\simplify{{c2}x ({a1}x^3+{b1}x^2+{c1}x+{d1})+{d2}({a1}x^3+{b1}x^2+{c1}x+{d1})} \\\\ &amp;\\,= \\simplify[all,!collectNumbers, !cancelTerms]{{c2*a1}x^4+{c2*b1}x^3+{c2*c1}x^2+{c2*d1}x +{d2*a1}x^3+{d2*b1}x^2+{d2*c1}x+{d2*d1}} \\\\ &amp;\\,=&nbsp;\\simplify{{c2*a1}x^4+{c2*b1}x^3+{c2*c1}x^2+{c2*d1}x +{d2*a1}x^3+{d2*b1}x^2+{d2*c1}x+{d2*d1}} \\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..4)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "b1", "c1", "d1", "c2", "d2"], "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": "{c2*a1}x^4+{c2*b1+d2*a1}x^3+{c2*c1+d2*b1}x^2+{c2*d1+d2*c1}x+{d1*d2}", "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", "type": "question", "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/"}]}