// Numbas version: finer_feedback_settings {"name": "Polynomials: Expand and Simplify 1b", "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 1b", "tags": [], "metadata": {"description": "
Multiplying a linear expression of the form $mx+c$ by a quadratic expression of the form $ax^2+bx+c$.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Given a function \\[ f(x) = \\simplify{{a1}x^2+{b1}x+{c1}},\\]
\ncalculate \\[(\\simplify{{m}x+{n}})\\left[f(x)\\right].\\]
", "advice": "Substituting the function $f(x) = \\simplify{{a1}x^2+{b1}x+{c1}}$ into $(\\simplify{{m}x+{n}})\\left[f(x)\\right]$:
\n\\[ \\begin{split} (\\simplify{{m}x+{n}})\\left[f(x)\\right] &\\,= \\simplify{({m}x+{n})({a1}x^2+{b1}x+{c1})} \\\\ &\\,= \\simplify[all]{{m}x({a1}x^2+{b1}x+{c1}) + {n}({a1}x^2+{b1}x+{c1})}. \\end{split}\\]
\nMultiplying out the brackets and collecting similar terms:
\n\\[ \\begin{split}\\simplify[all]{{m}x({a1}x^2+{b1}x+{c1}) + {n}({a1}x^2+{b1}x+{c1})} &\\,= \\simplify[all,!cancelTerms, !collectNumbers]{{m*a1}x^3+{m*b1}x^2+{m*c1}x + {n*a1}x^2+{n*b1}x+{n*c1}} \\\\ &\\,= \\simplify{{m*a1}x^3+{m*b1+n*a1}x^2+{m*c1+n*b1}x+{n*c1}}\\end{split} \\]
", "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..4)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(-3..3 except [0,m,-m,-1,1])", "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": "{m*a1}x^3+{m*b1+n*a1}x^2+{m*c1+n*b1}x+{n*c1}", "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": "`+-$n`?*x^3+`+-$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/"}]}