// Numbas version: exam_results_page_options
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\\left(\\simplify{{a1}x^2+{b1}x+{c1}}\\right)\\left(\\simplify{{a2-a1}x^2+{b2-b1}x+{c2-c1}}\\right) \\\\ &amp;\\,= \\simplify{{a1}x^2({a2-a1}x^2+{b2-b1}x+{c2-c1})+{b1}x({a2-a1}x^2+{b2-b1}x+{c2-c1})+{c1}({a2-a1}x^2+{b2-b1}x+{c2-c1})}. \\end{split} \\]</p>\n<p>We can now expand each of the brackets and collect similar terms:</p>\n<p>\\[ \\begin{split}&nbsp;f(x)\\left[g(x)-f(x)\\right] &amp;\\,=&nbsp;\\simplify{{a1}x^2({a2-a1}x^2+{b2-b1}x+{c2-c1})+{b1}x({a2-a1}x^2+{b2-b1}x+{c2-c1})+{c1}({a2-a1}x^2+{b2-b1}x+{c2-c1})} \\\\ &amp;\\,= \\simplify[all, !cancelTerms, !collectNumbers]{{a1*(a2-a1)}x^4+{a1*(b2-b1)}x^3+{a1*(c2-c1)}x^2 +&nbsp;{b1*(a2-a1)}x^3+{b1*(b2-b1)}x^2+{b1*(c2-c1)}x +&nbsp;{c1*(a2-a1)}x^2+{c1*(b2-b1)}x+{c1*(c2-c1)}} \\\\ &amp;\\,= \\simplify{{a1*da}x^4+{a1*db+da*b1}x^3+{a1*dc+da*c1+b1*db}x^2+{b1*dc+db*c1}x+{c1*dc}} \\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}, "da": {"name": "da", "group": "Ungrouped variables", "definition": "a2-a1", "description": "", "templateType": "anything", "can_override": false}, "db": {"name": "db", "group": "Ungrouped variables", "definition": "b2-b1", "description": "", "templateType": "anything", "can_override": false}, "dc": {"name": "dc", "group": "Ungrouped variables", "definition": "c2-c1", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "da>0", "maxRuns": 100}, "ungrouped_variables": ["a1", "a2", "b1", "b2", "c1", "c2", "m", "n", "da", "db", "dc"], "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*da}x^4+{a1*db+da*b1}x^3+{a1*dc+da*c1+b1*db}x^2+{b1*dc+db*c1}x+{c1*dc}", "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/"}]}