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{"name": "Polynomials: Expand and Simplify 1c", "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 1c", "tags": [], "metadata": {"description": "<p>Calculating a quartic polynomial by squaring a quadratic expression of the form $ax^2+bx+c$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Given&nbsp;a function \\[ f(x) = \\simplify{{a}x^2+{b}x+{c}},\\]</p>\n<p>calculate&nbsp;\\[ \\left[ f(x) \\right]^2 .\\]</p>", "advice": "<p>&nbsp;When calculating $\\left[f(x)\\right]^2$, it is important to remember that this means we are multiplying the <em>entire function</em> by itself, and not simply squaring each term.</p>\n<p>So,</p>\n<p>\\[ \\begin{split} \\left[f(x)\\right]^2 &amp;\\,= (\\simplify{{a}x^2+{b}x+{c}})(\\simplify{{a}x^2+{b}x+{c}}) \\\\ &amp;\\,= \\simplify{{a}x^2 ({a}x^2+{b}x+{c}) +{b}x({a}x^2+{b}x+{c})+{c}({a}x^2+{b}x+{c})} \\end{split} \\]</p>\n<p>Multiplying out the brackets and collecting similar terms:</p>\n<p>\\[ \\begin{split} \\left[f(x)\\right]^2 &amp;\\,= \\simplify{{a}x^2 ({a}x^2+{b}x+{c}) +{b}x({a}x^2+{b}x+{c})+{c}({a}x^2+{b}x+{c})} \\\\ &amp;\\,= \\simplify[all,!cancelTerms, !collectNumbers]{{a^2}x^4+{a*b}x^3+{a*c}x^2+{b*a}x^3+{b^2}x^2+{b*c}x+{c*a}x^2+{c*b}x+{c^2}}\\\\ &amp;\\,= \\simplify{{a^2}x^4+{a*b}x^3+{a*c}x^2+{b*a}x^3+{b^2}x^2+{b*c}x+{c*a}x^2+{c*b}x+{c^2}}\\end{split} \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-10..10 except b)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c"], "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": "{a^2}x^4+{2*a*b}x^3+{2*a*c+b^2}x^2+{2*b*c}x+{c^2}", "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/"}]}