// Numbas version: finer_feedback_settings
{"name": "Cubics: Finding a Quadratic Factor 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Cubics: Finding a Quadratic Factor 1", "tags": [], "metadata": {"description": "<p>Given a cubic expression and a linear factor, calculate the remaining quadratic factor.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p class=\"p1\">The cubic expression \\[ \\simplify{{m*a1}x^3+{m*b1+n*a1}x^2+{m*c1+n*b1}x+{n*c1}} \\]&nbsp;can be factorised into a product of a linear and quadratic factor.&nbsp;If the linear factor is $\\simplify{{m}x+{n}}$, find the remaining quadratic factor.</p>", "advice": "<p>If the cubic expression \\[ \\simplify{{m*a1}x^3+{m*b1+n*a1}x^2+{m*c1+n*b1}x+{n*c1}} \\]&nbsp;can be factorised into a product of a linear and quadratic factor, and we know that the linear factor is $\\simplify{{m}x+{n}}$, we can start by taking a generalised quadratic in the form $ax^2+bx+c$ as the quadratic factor:</p>\n<p>\\[ \\begin{split} \\simplify{{m*a1}x^3+{m*b1+n*a1}x^2+{m*c1+n*b1}x+{n*c1}} &amp;\\,= (\\simplify{{m}x+{n}})(ax^2+bx+c) \\\\ &amp;\\,= \\simplify{{m}x(a*x^2+b*x+c) + {n}(a*x^2+b*x+c)}. \\end{split} \\]</p>\n<p>If we now expand the brackets and collect similar terms, we can compare coefficients to find the values of $a$, $b$, and $c$:</p>\n<p>\\[ \\begin{split} \\simplify{{m*a1}x^3+{m*b1+n*a1}x^2+{m*c1+n*b1}x+{n*c1}} &amp;\\,= \\simplify{{m}x(a*x^2+b*x+c) + {n}(a*x^2+b*x+c)} \\\\ &amp;\\,=\\simplify[all,!collectNumbers, !cancelTerms]{{m}a x^3 + {m}b x^2 +{m} c x + {n}a x^2 + {n} b x + {n} c} \\\\ &amp;\\,=\\simplify{{m}a x^3 + ({m}b+{n}a)x^2+({m}c+{n}b)x +{n}c }.\\end{split} \\]</p>\n<p>By comparing the coefficients of each side of the equation we can create a set of simultaneous equations to find $a$, $b$, and $c$:</p>\n<p>\\[ \\begin{split} &amp; \\simplify{{m*a1}} &amp;\\,= \\simplify{{m}a} \\\\ &amp;\\simplify{{m*b1+n*a1}} &amp;\\,= \\simplify{{m}b+{n}a} \\\\ &amp;\\simplify{{m*c1+n*b1}} &amp;\\,=\\simplify{{m}c+{n}b} \\\\ &amp;\\simplify{{n*c1}} &amp;\\,=\\simplify{{n}c}. \\end{split} \\]</p>\n<p>Hence, \\[ a=\\var{a1},\\quad b=\\var{b1},\\quad c=\\var{c1}.\\]</p>\n<p>Therefore,&nbsp;\\[ \\simplify{{m*a1}x^3+{m*b1+n*a1}x^2+{m*c1+n*b1}x+{n*c1}} = (\\simplify{{m}x+{n}})(\\simplify{{a1}x^2+{b1}x+{c1}}) \\]</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,-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": "{a1}x^2+{b1}x+{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, "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/"}]}