// Numbas version: finer_feedback_settings
{"name": "Unit 1 Test", "metadata": {"description": "<p>Unit 1: Sequences and Functions</p>", "licence": "None specified"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Cubics: Finding Linear Factors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Factorise a cubic expression into the product of three linear factors of the form $mx+c$, where one of the factors is given.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Factorise the following cubic expression into a product of linear factors, where one of the factors is $\\simplify{{b1}x+{c1}}$:</p>\n<p>\\[\\simplify{{b1*b2*b3}x^3+{c1*b2*b3+c2*b1*b3+c3*b1*b2}x^2+{b1*c2*c3+b2*c1*c3+b3*c1*c2}x+{c1*c2*c3}} \\]</p>", "advice": "<p>If the cubic expression</p>\n<p>\\[\\simplify{{b1*b2*b3}x^3+{c1*b2*b3+c2*b1*b3+c3*b1*b2}x^2+{b1*c2*c3+b2*c1*c3+b3*c1*c2}x+{c1*c2*c3}} \\]</p>\n<p>can be factorised into a product of linear factors, and we know that&nbsp;one factor is $\\simplify{{b1}x+{c1}}$, it is easiest to first find the remaining quadratic factor. We can then factorise this quadratic into two linear factors.</p>\n<p>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{{b1*b2*b3}x^3+{c1*b2*b3+c2*b1*b3+c3*b1*b2}x^2+{b1*c2*c3+b2*c1*c3+b3*c1*c2}x+{c1*c2*c3}}&nbsp; &amp;\\,= (\\simplify{{b1}x+{c1}})(ax^2+bx+c) \\\\ &amp;\\,= \\simplify{{b1}x(a*x^2+b*x+c) + {c1}(a*x^2+b*x+c)}. \\end{split}&nbsp;\\]</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{{b1*b2*b3}x^3+{c1*b2*b3+c2*b1*b3+c3*b1*b2}x^2+{b1*c2*c3+b2*c1*c3+b3*c1*c2}x+{c1*c2*c3}}&nbsp;&amp;\\,= \\simplify{{b1}x(a*x^2+b*x+c) + {c1}(a*x^2+b*x+c)} \\\\ &amp;\\,=\\simplify[all,!collectNumbers, !cancelTerms]{{b1}a x^3 + {b1}b x^2 +{b1} c x + {c2}a x^2 + {c2} b x + {c2} c} \\\\ &amp;\\,=\\simplify{{b1}a x^3 + ({b1}b+{c1}a)x^2+({b1}c+{c1}b)x +{c1}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{{b1*b2*b3}} &amp;\\,= \\simplify{{b1}a} \\\\ &amp;\\simplify{{c1*b2*b3+c2*b1*b3+c3*b1*b2}} &amp;\\,= \\simplify{{b1}b+{c1}a} \\\\ &amp;\\simplify{{b1*c2*c3+b2*c1*c3+b3*c1*c2}} &amp;\\,=\\simplify{{b1}c+{c1}b} \\\\ &amp;\\simplify{{c1*c2*c3}} &amp;\\,=\\simplify{{c1}c}. \\end{split} \\]</p>\n<p>Hence, \\[ a=\\var{b2*b3},\\quad b=\\var{b1},\\quad c=\\var{c1}.\\]</p>\n<p>Therefore,&nbsp;\\[\\simplify{{b1*b2*b3}x^3+{c1*b2*b3+c2*b1*b3+c3*b1*b2}x^2+{b1*c2*c3+b2*c1*c3+b3*c1*c2}x+{c1*c2*c3}} = (\\simplify{{b1}x+{c1}})(\\simplify{{b2*b3}x^2+{c2*b3+b2*c3}x+{c2*c3}}). \\]</p>\n<p>Finally, we need to factorise the quadratic $\\simplify{{b2*b3}x^2+{c2*b3+b2*c3}x+{c2*c3}}$:</p>\n<p>\\[&nbsp;\\simplify{{b2*b3}x^2+{c2*b3+b2*c3}x+{c2*c3}}=(\\simplify{{b2}x+{c2}})(\\simplify{{b3}x+{c3}}).\\]</p>\n<p>Thus, the cubic expression as a product of three linear factors is</p>\n<p>\\[\\simplify{{b1*b2*b3}x^3+{c1*b2*b3+c2*b1*b3+c3*b1*b2}x^2+{b1*c2*c3+b2*c1*c3+b3*c1*c2}x+{c1*c2*c3}}=(\\simplify{{b1}x+{c1}})(\\simplify{{b2}x+{c2}})(\\simplify{{b3}x+{c3}}). \\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,-1])", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "c3": {"name": "c3", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,c2])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["b1", "c1", "b2", "c2", "b3", "c3"], "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": "({b1}x+{c1})({b2}x+{c2})({b3}x+{c3})", "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": "($n`?*x+`+-$n)($n`?*x+`+-$n)($n`?*x+`+-$n)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "typeendtoleave": false, "startpassword": "", "autoSubmit": true, "allowAttemptDownload": true, "downloadEncryptionKey": "25", "showresultspage": "oncompletion"}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"enterreviewmodeimmediately": true, "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showpartfeedbackmessageswhen": "always", "showexpectedanswerswhen": "inreview", "showadvicewhen": "never", "activatereviewmode": "oncompletion", "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "end_message": "", "results_options": {"printquestions": true, "printadvice": true}, "feedbackmessages": [], "reviewshowexpectedanswer": true, "showanswerstate": true, "reviewshowfeedback": true, "showactualmark": true, "showtotalmark": true, "reviewshowscore": true, "reviewshowadvice": false}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Stephen  Abreu", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1580/"}], "extensions": [], "custom_part_types": [], "resources": []}