// Numbas version: exam_results_page_options
{"name": "Partial Fractions: Proper Fractions 3b", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Partial Fractions: Proper Fractions 3b", "tags": [], "metadata": {"description": "<p>Rewrite the expression $\\frac{cx+d}{x^2+mx+n}$ as partial fractions in the form $\\frac{A}{x+a}+\\frac{B}{x+b}$, where the quadratic $x^2+mx+n=(x+a)(x+b)$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Rewrite the following expression as partial fractions:</p>\n<p>\\[ \\simplify{({c}x+{d})/(x^2+{a+b}x+{a*b})} .\\]</p>", "advice": "<p>To express\\[ \\simplify{({c}x+{d})/(x^2+{a+b}x+{a*b})} \\] as partial fractions, we must first factorise the denominator into linear factors:</p>\n<p>\\[ \\simplify{({c}x+{d})/(x^2+{a+b}x+{a*b})}=\\simplify{({c}x+{d})/((x+{a})(x+{b}))}.\\]</p>\n<p>Now the denominator is factorised, we want to set this equal to the sum of 2 fractions with denominators $\\simplify{x+{a}}$ and $\\simplify{x+{b}}$. Since these are both distinct linear factors, this tells us that the numerators will be constants, which we will call $A$ and $B$:</p>\n<p>\\[ \\simplify{({c}x+{d})/((x+{a})(x+{b}))} = \\simplify{A/(x+{a}) + B/(x+{b})}.\\]</p>\n<p>To find the values of $A$ and $B$, we want to multiply this equation by the denominator of the left-hand side. This gives</p>\n<p>\\[ \\simplify{{c}x+{d}=A(x+{b})+B(x+{a})}.\\]</p>\n<p>There are 2 methods of finding $A$ and $B$. The first is to choose suitable values for $x$ which will eliminate one of the terms, and the other is to compare the coefficients of each side of the equation. We will cover both methods here.</p>\n<p><strong>Method 1:</strong></p>\n<p>To find $A$, we can eliminate $B$ by setting $\\simplify{x={-a}}$:</p>\n<p>\\[ \\simplify{{d-c*a}=A{b-a}} \\implies \\simplify[fractionNumbers]{A={(d-c*a)/(b-a)}}.\\]</p>\n<p>Similarly, to find B, we can eliminate $A$ by setting $\\simplify{x={-b}}$:</p>\n<p>\\[ \\simplify{{d-c*b}=B{a-b}} \\implies \\simplify[fractionNumbers]{B={(d-c*b)/(a-b)}}.\\]</p>\n<p>Therefore,&nbsp;</p>\n<p>{check}</p>\n<p><strong>Method 2:</strong></p>\n<p>By comparing the coefficients of the $x$-terms and the constant terms we can form a pair of simultaneous equations to find $A$ and $B$.</p>\n<p>\\[ \\begin{split} \\simplify{{c}x+{d}} &amp;\\,= \\simplify{A(x+{b})+B(x+{a})} \\\\ &amp;\\,= \\simplify{A*x+{b}A+B*x +{a}B} \\\\ &amp;\\,=\\simplify{(A+B)x +{b}A+{a}B} . \\end{split} \\]</p>\n<p>\\[ \\begin{split}&amp;(x):\\quad \\var{c} &amp;\\,= \\simplify{A+B} \\\\ &amp;(c):\\quad \\var{d} &amp;\\,= \\simplify{{b}A+{a}B} .\\end{split} \\]</p>\n<p>Hence,</p>\n<p>\\[A=\\simplify[fractionNumbers]{{Asol}},\\,B=\\simplify[fractionNumbers]{{Bsol}}, \\]</p>\n<p>and</p>\n<p>{check}</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"Asol": {"name": "Asol", "group": "Ungrouped variables", "definition": "(d-c*a)/(b-a)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-9..9 except a)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "Bsol": {"name": "Bsol", "group": "Ungrouped variables", "definition": "(d-c*b)/(a-b)", "description": "", "templateType": "anything", "can_override": false}, "check": {"name": "check", "group": "Ungrouped variables", "definition": "if(Asol=round(Asol),'{Sol1}','{Sol2}')", "description": "", "templateType": "anything", "can_override": false}, "Sol1": {"name": "Sol1", "group": "Ungrouped variables", "definition": "\"<p>\\\\[&nbsp;\\\\simplify{({c}x+{d})/(x^2+{a+b}x+{a*b})} = \\\\simplify[all,fractionNumbers]{{Asol}/(x+{a})+{Bsol}/(x+{b})}.\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "Sol2": {"name": "Sol2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[&nbsp;\\\\simplify{({c}x+{d})/(x^2+{a+b}x+{a*b})} = \\\\simplify[all,fractionNumbers]{{d-c*a}/(({b-a})(x+{a}))+{d-c*b}/(({a-b})(x+{b}))}.\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-9..9 except[c*a,c*b])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "gcd(c,d)=1", "maxRuns": 100}, "ungrouped_variables": ["Asol", "b", "a", "c", "d", "Bsol", "check", "Sol1", "Sol2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n<p>[[0]]</p>", "gaps": [{"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": "{(d-c*a)}/(({b-a})(x+{a}))+{d-c*b}/(({a-b})(x+{b}))", "answerSimplification": "all", "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": "`+-$n`?/($n`?*($n`?x+`+-$n`?))+`+-$n`?/($n`?*($n`?x+`+-$n`?)) `| `+-$n/`+-$n*(`+-$n`?/($n`?*($n`?x+`+-$n`?))+`+-$n`?/($n`?*($n`?x+`+-$n`?)))", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "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/"}]}