// Numbas version: exam_results_page_options {"name": "Maths Support: Partial fractions", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 0.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Partial Fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d"], "tags": ["rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"vsetrangepoints": 5, "prompt": "

Factorise $\\simplify{ x^2 + {b+d}x + {b*d} }$

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Express $\\frac{\\simplify{{a+c}*x+{a*d+c*b}}}{\\simplify{ x^2 + {b+d}x + {b*d} }}$ in partial fractions.

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Splitting an algebraic fraction into partial fractions

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rebelmaths

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We use partial fractions to find $A$ and $B$ such that: 
\\[ \\simplify[std]{({a*a2+c*a1}*x+{c*b+a*d})/(({a1}x +{b})*({a2}x+{d}))} \\;\\;\\;=\\simplify[std]{ A/({a1}x+{b})+B/({a2}x+{d})}\\]

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Multiplying both sides of the equation by $\\displaystyle \\simplify[std]{1/( ({a1}x+{b})({a2}x+{d}) )}\\;\\;$ we obtain:

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$\\simplify[std]{A*({a2}x+{d})+B*({a1}x+{b}) = {a*a2+c*a1}*x+{a*d+c*b}} \\Rightarrow \\simplify[std]{({a2}A+{a1}B)*x+{d}*A+{b}*B={a*a2+c*a1}*x+{a*d+c*b}}$

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Identifying coefficients:

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Constant term: $\\simplify[std]{ {d}*A+{b}*B={a*d+c*b} }$

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Coefficent $x$: $ \\simplify[std]{ {a2}A+{a1}B = {a*a2+c*a1} }$ 

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On solving these equations we obtain $A = \\var{a}$ and $B=\\var{c}$

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Which gives:\\[ \\simplify[std]{({a*a2+c*a1}*x+{c*b+a*d})/(({a1}x +{b})*({a2}x+{d}))}\\;\\;= \\simplify[std]{{a}/({a1}x+{b})+{c}/({a2}x+{d})}\\]

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Split \\[\\simplify{({a*a2 +  c*a1} * x + {a * d +  c * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))}\\] into partial fractions.

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Input the partial fractions here: [[0]].

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\n ", "gaps": [{"notallowed": {"message": "

Input as the sum of partial fractions.

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5/08/2012:

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Added tags.

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Added description.

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Changed to two questions, for the numerator and denomimator, rather than one as difficult to trap student input for this example. Still some ambiguity however.

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12/08/2012:

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Back to one input of a fraction and trapped input in Forbidden Strings.

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Used the except feature of ranges to get non-degenerate examples.

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Checked calculation.OK.

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Improved display in content areas.

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Split $\\displaystyle \\frac{ax+b}{(cx + d)(px+q)}$ into partial fractions.

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