Practice partial fractions. Preparing for integration using partial fractions.
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n\n
\n ", "advice": "
a)
\nWe 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})}\\]
Dividing both sides of the equation by $\\displaystyle \\simplify[std]{1/( ({a1}x+{b})({a2}x+{d}) )}\\;\\;$ we obtain:
\n$\\simplify[std]{A*({a2}x+{d})+B*({a1}x+{b}) = {a*a2+c*a1}*x+{a*d+c*b}} \\Rightarrow \\simplify[std]{({a2}A+{a1}B)*x+{d}*A+{b}*B={a*a2+c*a1}*x+{a*d+c*b}}$
\nIdentifying coefficients:
\nConstant term: $\\simplify[std]{ {d}*A+{b}*B={a*d+c*b} }$
\nCoefficent $x$: $ \\simplify[std]{ {a2}A+{a1}B = {a*a2+c*a1} }$
\nOn solving these equations we obtain $A = \\var{a}$ and $B=\\var{c}$
\nWhich 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})}\\]
\n\nb)
\nWe 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_})}\\]
Dividing both sides of the equation by $\\displaystyle \\simplify[std]{1/( ({a1_}x+{b_})({a2_}x+{d_}) )}\\;\\;$ we obtain:
\n$\\simplify[std]{A*({a2_}x+{d_})+B*({a1_}x+{b_}) = {a_*a2_+c_*a1_}*x+{a_*d_+c_*b_}} \\Rightarrow \\simplify[std]{({a2_}A+{a1_}B)*x+{d_}*A+{b_}*B={a_*a2_+c_*a1_}*x+{a_*d_+c_*b_}}$
\nIdentifying coefficients:
\nConstant term: $\\simplify[std]{ {d_}*A+{b_}*B={a_*d_+c_*b_} }$
\nCoefficent $x$: $ \\simplify[std]{ {a2_}A+{a1_}B = {a_*a2_+c_*a1_} }$
\nOn solving these equations we obtain $A = \\var{a_}$ and $B=\\var{c_}$
\nWhich 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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\nInput the partial fractions here: [[0]].
\n\n
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Input as the sum of partial fractions.
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\nInput the partial fractions here: [[0]].
\n\n
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Input as the sum of partial fractions.
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\\[ \\simplify[std]{({a*a1}*x+{a*b+c})/(({a1}x +{b})^2)} = \\simplify[std]{ A/({a1}x+{b})+B/({a1}x+{b})^2}\\]
Multiplying both sides of the equation by $\\displaystyle \\simplify[std]{1/( ({a1}x+{b})^2)}$ we obtain:
\n$\\simplify[std]{A*({a1}x+{b})+B = {a*a1}*x+{a*b+c}} \\Rightarrow \\simplify[std]{({a1}A)*x+{b}*A+B={a*a1}*x+{a*b+c}}$
\nIdentifying coefficients:
\nConstant term: $\\simplify[std]{ {b}*A+B={a*b+c} }$
\nCoefficent $x$: $ \\simplify[std]{ {a1}A = {a*a1} }$
\nOn solving these equations we obtain $A = \\var{a}$ and $B=\\var{c}$
\nWhich gives:\\[ \\simplify[std]{({a*a1}*x+{a*b+c})/(({a1}x +{b})^2)} = \\simplify[std]{{a}/({a1}x+{b})+{c}/({a1}x+{b})^2}.\\]
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", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(-9..9 except [0,-a])", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-9..9 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "nb": {"definition": "if(c<0,'taking away','adding')", "templateType": "anything", "group": "Ungrouped variables", "name": "nb", "description": ""}, "a1": {"definition": "random(1..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "s1": {"definition": "if(c<0,-1,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}}, "metadata": {"description": "Split $\\displaystyle \\frac{ax+b}{(cx + d)^2}$ into partial fractions.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question"}, {"name": "Partial Fractions: Repeated Factor", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "variablesTest": {"maxRuns": "100000", "condition": "b1+b3=0\n"}, "statement": "\n\n
\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": "", "templateType": "anything", "definition": "if(c_<0,-1,1)", "group": "Ungrouped variables", "name": "s1_"}, "b2": {"description": "", "templateType": "anything", "definition": "random(-5..5 except 0)", "group": "Ungrouped variables", "name": "b2"}, "c_": {"description": "", "templateType": "anything", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "name": "c_"}, "c1": {"description": "", "templateType": "anything", "definition": "random(-5..5 except 0)", "group": "Ungrouped variables", "name": "c1"}, "c": {"description": "", "templateType": "anything", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "name": "c"}, "p1": {"description": "", "templateType": "anything", "definition": "random(-5..5 except 0)", "group": "Ungrouped variables", "name": "p1"}, "b3": {"description": "", "templateType": "anything", "definition": "random(-5..5 except 0)", "group": "Ungrouped variables", "name": "b3"}, "c2": {"description": "", "templateType": "anything", "definition": "random(-5..5 except 0)", "group": "Ungrouped variables", "name": "c2"}, "a": {"description": "", "templateType": "anything", "definition": "random(1..9)", "group": "Ungrouped variables", "name": "a"}, "a_": {"description": "", "templateType": "anything", "definition": "random(1..9)", "group": "Ungrouped variables", "name": "a_"}, "s1": {"description": "", "templateType": "anything", "definition": "if(c<0,-1,1)", "group": "Ungrouped variables", "name": "s1"}, "d_": {"description": "", "templateType": "anything", "definition": "1", "group": "Ungrouped variables", "name": "d_"}, "d": {"description": "", "templateType": "anything", "definition": "1", "group": "Ungrouped variables", "name": "d"}, "q1": {"description": "", "templateType": "anything", "definition": "random(-5..5 except 0 except p1)", "group": "Ungrouped variables", "name": "q1"}, "a3": {"description": "", "templateType": "anything", "definition": "random(1..9)", "group": "Ungrouped variables", "name": "a3"}, "nb_": {"description": "", "templateType": "anything", "definition": "if(c_<0,'taking away','adding')", "group": "Ungrouped variables", "name": "nb_"}, "a1": {"description": "", "templateType": "anything", "definition": "random(1..9)", "group": "Ungrouped variables", "name": "a1"}, "p": {"description": "", "templateType": "anything", "definition": "random(-5..5 except 0)", "group": "Ungrouped variables", "name": "p"}, "a2_": {"description": "", "templateType": "anything", "definition": "1", "group": "Ungrouped variables", "name": "a2_"}, "a1_": {"description": "", "templateType": "anything", "definition": "1", "group": "Ungrouped variables", "name": "a1_"}, "q": {"description": "", "templateType": "anything", "definition": "random(-5..5 except 0 except p)", "group": "Ungrouped variables", "name": "q"}, "b1": {"description": "", "templateType": "anything", "definition": "random(-5..5 except 0)", "group": "Ungrouped variables", "name": "b1"}, "b": {"description": "", "templateType": "anything", "definition": "random(-9..9 except 0 except a)", "group": "Ungrouped variables", "name": "b"}, "b_": {"description": "", "templateType": "anything", "definition": "random(-9..9 except 0)", "group": "Ungrouped variables", "name": "b_"}}, "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": ""}, "variable_groups": [], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "tags": ["algebra", "Algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "ungrouped_variables": ["a", "c", "b", "nb", "s1", "a_", "c_", "b_", "nb_", "a1_", "a2_", "s1_", "new", "a1", "a2", "a3", "d", "d_", "b1", "b2", "b3", "p", "q", "c1", "c2", "c3", "p1", "q1"], "parts": [{"type": "gapfill", "scripts": {}, "extendBaseMarkingAlgorithm": true, "useCustomName": false, "adaptiveMarkingPenalty": 0, "customName": "", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "unitTests": [], "prompt": "
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.
\nInput the partial fractions here: [[0]].
\n\n
", "customMarkingAlgorithm": "", "gaps": [{"type": "jme", "scripts": {}, "useCustomName": false, "customName": "", "notallowed": {"showStrings": false, "message": "
Input as the sum of partial fractions.
", "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": "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.
\nInput the partial fractions here: [[0]].
\n\n
", "customMarkingAlgorithm": "", "gaps": [{"type": "jme", "scripts": {}, "useCustomName": false, "customName": "", "notallowed": {"showStrings": false, "message": "
Input as the sum of partial fractions.
", "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": "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.
\nInput the partial fractions here: [[0]].
\n\n
", "customMarkingAlgorithm": "", "gaps": [{"type": "jme", "scripts": {}, "useCustomName": false, "customName": "", "notallowed": {"showStrings": false, "message": "
Input as the sum of partial fractions.
", "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": ""}, "advice": "a)
\nWe use partial fractions to find $A$, $B$ and $C$ such that:
$\\simplify{({a1+a3}x^2+{a1*a+a1*b+a2+2*a*a3} * x + {a1*a*b + a2*b + a3*a^2})/ ((x + {a})^2 * (x + {b}))} \\;\\;\\;=\\simplify{A/(x+{a})+B/(x+{a})^2+C/(x+{b})}$
Dividing both sides of the equation by $\\displaystyle \\simplify[std]{1/( (x+{a})^2(x+{b}) )}\\;\\;$ we obtain:
\n$ \\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}}$
\n$\\Rightarrow \\simplify[std]{(A+C)x^2+({a+b}A+B+{2a}C)x+({a*b}A+{b}B+{a*a}C)={a1+a3}*x^2+{a1*a+a1*b+a2+2*a*a3}*x + {a1*a*b + a2*b + a3*a^2}}$
\nIdentifying coefficients:
\nCoefficient $x^2$: $\\simplify[std]{A+C={a1+a3} }$
\nCoefficent $x$: $ \\simplify[std]{ {a+b}A+B+{2a}C = {a1*a+a1*b+a2+2*a*a3} }$
\nConstant term: $\\simplify{{a*b}A+{b}B+{a*a}C ={a1*a*b + a2*b + a3*a^2}}$
\nOn solving these equations we obtain $A = \\var{a1}$, $B=\\var{a2}$ and $C=\\var{a3}$
\nWhich gives:$\\simplify{({a1+a3}x^2+{a1*a+a1*b+a2+2*a*a3} * x + {a1*a*b + a2*b + a3*a^2})/ ((x + {a})^2 * (x + {b}))} \\;\\;\\;=\\simplify{{a1}/(x+{a})+{a2}/(x+{a})^2+{a3}/(x+{b})}$
\n\nApply same method to solve b) and c)
", "type": "question"}, {"name": "Partial Fractions - quadratic term in denominator", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}], "functions": {}, "ungrouped_variables": ["coeff0", "coeff1", "a", "a1", "a2", "a3", "bcoeff0", "bcoeff1", "b", "b1", "b2", "b3", "ccoeff0", "ccoeff1", "c", "c1", "c2", "c3"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "Suppose we want to express
\n$\\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})/((x^2+{coeff1}x+{coeff0})(x-{a}))}$
\nas the sum of its partial fractions.
\nWhen the denominator contains a quadratic factor we have to consider the possibilty that the numebrator can contain a term in $x$. This is because if it did, the numerator would still be of lower degree than the denominator - this would be a proper fraction. So we write
\n$\\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})/((x^2+{coeff1}x+{coeff0})(x-{a}))}=\\simplify{(Ax+B)/(x^2+{coeff1}x+{coeff0})+c/(x+{a})}$
\nWe multiply both sides by $\\simplify{((x^2+{coeff1}x+{coeff0})(x-{a}))}$ to give $\\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})}=\\simplify{(Ax+B)(x+{a})+C(x^2+{coeff1}x+{coeff0})}$
\nBy evaluating both sides at $\\simplify{x=-{a}}$ we can find C$=\\var{a3}$
\nThen by comparing coefficients we find A$=\\var{a1}$ and B$=\\var{a2}$
\n\nApply the same method for b) and c)
", "rulesets": {}, "parts": [{"prompt": "Express the following as a sum of partial fractions
\n$\\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})/((x^2+{coeff1}x+{coeff0})(x-{a}))}$
\n[[0]]
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\n$\\simplify{({b1+b3}x^2+{b2-b*b1+b3*bcoeff1}x+{b3*bcoeff0-b2*b})/((x^2+{bcoeff1}x+{bcoeff0})(x-{b}))}$
\n[[0]]
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\n$\\simplify{({c1+c3}x^2+{c2-c*c1+c3*ccoeff1}x+{c3*ccoeff0-c2*c})/((x^2+{ccoeff1}x+{ccoeff0})(x-{c}))}$
\n[[0]]
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\n$\\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})/((x^2+{coeff1}x+{coeff0})(x-{a}))}$
\nas the sum of its partial fractions.
\nWhen the denominator contains a quadratic factor we have to consider the possibilty that the numebrator can contain a term in $x$. This is because if it did, the numerator would still be of lower degree than the denominator - this would be a proper fraction. So we write
\n$\\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})/((x^2+{coeff1}x+{coeff0})(x-{a}))}=\\simplify{(Ax+B)/(x^2+{coeff1}x+{coeff0})+c/(x+{a})}$
\nWe multiply both sides by $\\simplify{((x^2+{coeff1}x+{coeff0})(x-{a}))}$ to give $\\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})}=\\simplify{(Ax+B)(x+{a})+C(x^2+{coeff1}x+{coeff0})}$
\nBy evaluating both sides at $\\simplify{x=-{a}}$ we can find C$=\\var{a3}$
\nThen by comparing coefficients we find A$=\\var{a1}$ and B$=\\var{a2}$
\n\nApply the same method for b) and c)
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