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{"name": "Partial Fractions: Repeated Factor", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"maxRuns": "100000", "condition": "b1+b3=0\n"}, "statement": "\n    <p>&nbsp;</p>\n  <p>&nbsp;</p>\n    ", "variables": {"nb": {"description": "", "templateType": "anything", "definition": "if(c<0,'taking away','adding')", "group": "Ungrouped variables", "name": "nb"}, "c3": {"description": "", "templateType": "anything", "definition": "random(-5..5 except 0)", "group": "Ungrouped variables", "name": "c3"}, "new": {"description": "", "templateType": "anything", "definition": "1", "group": "Ungrouped variables", "name": "new"}, "a2": {"description": "", "templateType": "anything", "definition": "random(1..9)", "group": "Ungrouped variables", "name": "a2"}, "s1_": {"description": "", 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"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "unitTests": [], "prompt": "<p>Split \\[\\simplify{(({a1+a3})x^2+{a1*a+a1*b+a2+2*a*a3} * x + {a1*a*b + a2*b + a3*a^2})/ ((x + {a})^2 * (x + {b}))}\\] into partial fractions.</p>\n<p>Input the partial fractions here: [[0]].</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "customMarkingAlgorithm": "", "gaps": [{"type": "jme", "scripts": {}, "useCustomName": false, "customName": "", "notallowed": {"showStrings": false, "message": "<p>Input as the sum of partial fractions.</p>", "strings": [")(", ")*("], "partialCredit": 0}, "answer": "{a1} / (x + {a}) + {a2}/(x + {a})^2 + {a3} / (x + {b})", "checkVariableNames": false, "vsetRangePoints": 5, "unitTests": [], "vsetRange": [10, 11], "customMarkingAlgorithm": "", "marks": 2, "failureRate": 1, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "checkingType": "absdiff", "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "valuegenerators": [{"value": "", "name": "x"}], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 1e-05, "showCorrectAnswer": true, "showPreview": true}], "marks": 0, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": []}, {"type": "gapfill", "scripts": {}, "extendBaseMarkingAlgorithm": true, "useCustomName": false, "adaptiveMarkingPenalty": 0, "customName": "", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "unitTests": [], "prompt": "<p>Split \\[\\simplify{(({b1+b3})x^2+{b1*p+b1*q+b2+2*p*b3} * x + {b1*p*q + b2*q + b3*p^2})/ ((x + {p})^2 * (x + {q}))}\\] into partial fractions.</p>\n<p>Input the partial fractions here: [[0]].</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "customMarkingAlgorithm": "", "gaps": [{"type": "jme", "scripts": {}, "useCustomName": false, "customName": "", "notallowed": {"showStrings": false, "message": "<p>Input as the sum of partial fractions.</p>", "strings": [")(", ")*("], "partialCredit": 0}, "answer": "{b1} / (x + {p}) + {b2}/(x + {p})^2 + {b3} / (x + {q})", "checkVariableNames": false, "vsetRangePoints": 5, "unitTests": [], "vsetRange": [10, 11], "customMarkingAlgorithm": "", "marks": 2, "failureRate": 1, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "checkingType": "absdiff", "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "valuegenerators": [{"value": "", "name": "x"}], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 1e-05, "showCorrectAnswer": true, "showPreview": true}], "marks": 0, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": []}, {"type": "gapfill", "scripts": {}, "extendBaseMarkingAlgorithm": true, "useCustomName": false, "adaptiveMarkingPenalty": 0, "customName": "", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "unitTests": [], "prompt": "<p>Split \\[\\simplify{(({c1+c3})x^2+{c1*p1+c1*q1+c2+2*p1*c3} * x + {c1*p1*q1 + c2*q1 + c3*p1^2})/ ((x + {p1})^2 * (x + {q1}))}\\] into partial fractions.</p>\n<p>Input the partial fractions here: [[0]].</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "customMarkingAlgorithm": "", "gaps": [{"type": "jme", "scripts": {}, "useCustomName": false, "customName": "", "notallowed": {"showStrings": false, "message": "<p>Input as the sum of partial fractions.</p>", "strings": [")(", ")*("], "partialCredit": 0}, "answer": "{c1} / (x + {p1}) + {c2}/(x + {p1})^2 + {c3} / (x + {q1})", "checkVariableNames": false, "vsetRangePoints": 5, "unitTests": [], "vsetRange": [10, 11], "customMarkingAlgorithm": "", "marks": 2, "failureRate": 1, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "checkingType": "absdiff", "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "valuegenerators": [{"value": "", "name": "x"}], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 1e-05, "showCorrectAnswer": true, "showPreview": true}], "marks": 0, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": []}], "preamble": {"js": "", "css": ""}, "name": "Partial Fractions: Repeated Factor", "advice": "<p>a)</p>\n<p>We use partial fractions to find $A$, $B$ and $C$ such that:&nbsp;<br/>$\\simplify{({a1+a3}x^2+{a1*a+a1*b+a2+2*a*a3} * x + {a1*a*b + a2*b + a3*a^2})/ ((x + {a})^2 * (x + {b}))} \\;\\;\\;=\\simplify{A/(x+{a})+B/(x+{a})^2+C/(x+{b})}$</p>\n<p>Dividing&nbsp;both sides of the equation by $\\displaystyle \\simplify[std]{1/( (x+{a})^2(x+{b})&nbsp;)}\\;\\;$ we obtain:</p>\n<p>$ \\simplify{A(x+{a})(x+{b})+B(x+{b})+C(x+{a})^2 = {a1+a3}*x^2+{a1*a+a1*b+a2+2*a*a3}*x + {a1*a*b + a2*b + a3*a^2}}$</p>\n<p>$\\Rightarrow&nbsp;\\simplify[std]{(A+C)x^2+({a+b}A+B+{2a}C)x+({a*b}A+{b}B+{a*a}C)={a1+a3}*x^2+{a1*a+a1*b+a2+2*a*a3}*x + {a1*a*b + a2*b + a3*a^2}}$</p>\n<p><strong>Identifying coefficients:</strong></p>\n<p>Coefficient $x^2$: $\\simplify[std]{A+C={a1+a3}&nbsp;}$</p>\n<p>Coefficent $x$: $ \\simplify[std]{ {a+b}A+B+{2a}C&nbsp;= {a1*a+a1*b+a2+2*a*a3}&nbsp;}$&nbsp;</p>\n<p>Constant term: $\\simplify{{a*b}A+{b}B+{a*a}C ={a1*a*b + a2*b + a3*a^2}}$</p>\n<p>On solving these equations we obtain $A = \\var{a1}$, $B=\\var{a2}$ and $C=\\var{a3}$</p>\n<p>Which gives:$\\simplify{({a1+a3}x^2+{a1*a+a1*b+a2+2*a*a3} * x + {a1*a*b + a2*b + a3*a^2})/ ((x + {a})^2 * (x + {b}))} \\;\\;\\;=\\simplify{{a1}/(x+{a})+{a2}/(x+{a})^2+{a3}/(x+{b})}$</p>\n<p></p>\n<p>Apply same method to solve b) and c)</p>", "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}