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Quiz to test students on partial fractions.

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Resolve the following fraction into partial fractions.

\\[\\frac{\\simplify{{a+b+c}*x^2+{a*d+a*f+b*f+c*d}*x+{a*d*f}}}{\\simplify{x^3+{d+f}*x^2+{d*f}*x}}\\]

", "advice": "

Factorise the denominator by taking out the common factor of $x$:

\\[\\simplify{x^3+{d+f}*x^2+{d*f}*x}=x\\left(\\simplify{x^2+{d+f}*x+{d*f}}\\right)\\]

Now factorise the bracket:

\\[\\simplify{x^3+{d+f}*x^2+{d*f}*x}=x(\\simplify{x+{d}})(\\simplify{x+{f}})\\]

Therefore we have three linear factors and the partial fractions are of the form:

\\[\\frac{\\simplify{{a+b+c}*x^2+{a*d+a*f+b*f+c*d}*x+{a*d*f}}}{\\simplify{x^3+{d+f}*x^2+{d*f}*x}}=\\frac{A}{x}+\\frac{B}{\\simplify{x+{d}}}+\\frac{C}{\\simplify{x+{f}}}\\]

Comvert each fraction so it has denominiator $x(\\simplify{x+{d}})(\\simplify{x+{f}})$:

\\[\\frac{A(\\simplify{x+{d}})(\\simplify{x+{f}})}{x(\\simplify{x+{d}})(\\simplify{x+{f}})}+\\frac{Bx(\\simplify{x+{f}})}{x(\\simplify{x+{d}})(\\simplify{x+{f}})}+\\frac{Cx(\\simplify{x+{d}})}{x(\\simplify{x+{d}})(\\simplify{x+{f}})}\\]

Combine fractions into one:

\\[\\frac{A(\\simplify{x+{d}})(\\simplify{x+{f}})+Bx(\\simplify{x+{f}})+Cx(\\simplify{x+{d}})}{x(\\simplify{x+{d}})(\\simplify{x+{f}})}\\]

Now equate the numerators:

\\[\\simplify{{a+b+c}*x^2+{a*d+a*f+b*f+c*d}*x+{a*d*f}}=A(\\simplify{x+{d}})(\\simplify{x+{f}})+Bx(\\simplify{x+{f}})+Cx(\\simplify{x+{d}})\\]

Let $x=0$:

\\[\\var{a+b+c}\\times0^2+\\var{a*d+a*f+b*f+c*d}\\times0+\\var{a*d*f}=A(0+\\var{d})(0+\\var{f})+B(0)(0+\\var{f})+C(0)(0+\\var{d})\\]

Simpifying this gives $\\simplify{{a*d*f}}=\\var{d*f}A$; therefore $A=\\var{a}$.

Now let $x=\\var{-d}$:

\\[\\var{a+b+c}\\times\\var{-d}^2+\\var{a*d+a*f+b*f+c*d}\\times\\var{-d}+\\var{a*d*f}=A(\\var{-d}+\\var{d})(\\var{-d}+\\var{f})+B(\\var{-d})(\\var{-d}+\\var{f})+C(\\var{-d})(\\var{-d}+\\var{d})\\]

Simplifying this gives $\\var{b*d*(d-f)}=\\var{d*(d-f)}B$ so $B=\\var{b}$.

Now let $x=\\var{-f}$:

\\[\\var{a+b+c}\\times\\var{-f}^2+\\var{a*d+a*f+b*f+c*d}\\times\\var{d}+\\var{a*d*f}=A(\\var{-f}+\\var{d})(\\var{-f}+\\var{f})+B(\\var{-f})(\\var{-f}+\\var{f})+C(\\var{-f})(\\var{-f}+\\var{d})\\]

Simplifying this gives $\\var{c*f*(f-d)}=\\var{f*(f-d)}C$; therefore $C=\\var{c}$.

Substitute $A$, $B$ and $C$ to obtain the final answer:

\\[\\frac{\\var{a}}{x}+\\frac{\\var{b}}{\\simplify{x+{d}}}+\\frac{\\var{c}}{\\simplify{x+{f}}}\\]

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Fully factorise the denominator: $\\simplify{x^3+{d+f}*x^2+{d*f}*x}$

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First take out the common factor of $x$ from each term. What polynomial is left?

\\[\\simplify{x^3+{d+f}*x^2+{d*f}*x}=x\\big(\\quad ??? \\quad\\big)\\]

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Now factorise the quadratic $\\simplify{x^2+{d+f}*x+{d*f}}$.

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No powers required here.

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Therefore you can fully factorise in the form $x(x\\qquad)(x\\qquad)$

"}], "answer": "x*(x+{d})*(x+{f})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["^"], "showStrings": true, "partialCredit": 0, "message": "

No powers are required

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Select the correct form of the partial fraction:

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\\[\\frac{A}{x}+\\frac{B}{\\simplify{x+{d}}}+\\frac{C}{\\simplify{x+{f}}}\\]

", "

\\[\\frac{A}{x}+\\frac{B}{\\simplify{x+{d}}}+\\frac{C}{\\simplify{x+{f+1}}}\\]

", "

\\[\\frac{A}{x}+\\frac{B}{\\simplify{x+{d-1}}}+\\frac{C}{\\simplify{x+{f}}}\\]

", "

\\[\\frac{A}{x}+\\frac{B}{\\simplify{x+{d}}}+\\frac{C}{\\simplify{x-{f}}}\\]

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Combine fractions into one:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$A$	$+$	$B$	$+$	$C$	$=$	$A$[[0]]+$B$[[1]]+$C$[[2]]
$\\simplify{x}$		$\\simplify{x+{d}}$		$\\simplify{x+{f}}$		$x\\simplify{(x+{d})*(x+{f})}$

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We now equate the numerators of the original fraction and your fraction from step c), i.e.

\\[\\simplify{{a+b+c}*x^2+{a*d+a*f+b*f+c*d}*x+{a*d*f}}=A\\simplify{(x+{d})*(x+{f})}+Bx(\\simplify{x+{f}})+Cx(\\simplify{x+{d}})\\]

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Substitute $x=0$: [[0]] = [[1]]

Therefore $A=$ [[2]]

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Substitute $x=\\var{-d}$: [[0]] = [[1]]

Therefore $B=$ [[2]]

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Substitute $x=\\var{-f}$: [[0]] = [[1]]

Therefore $C=$ [[2]]

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Therefore

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\simplify{{a+b+c}x^2+{ad+af+bf+cd}x+{adf}}$	$=$	[[0]]	$+$	[[1]]	$+$	[[2]]
$\\simplify{x^3+{d+f}x^2+{df}*x}$		$\\simplify{x}$		$\\simplify{x+{d}}$		$\\simplify{x+{f}}$

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Resolve the following fraction into partial fractions.

\\[\\frac{\\simplify{{a+b}*x^2+{2*a*f+b*d+b*f+c}*x+{a*f^2+b*d*f+c*d}}}{\\simplify{(x+{d})*(x+{f})^2}}\\]

", "advice": "", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything"}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(-6..6 except [0,d])", "description": "", "templateType": "anything"}, "xmd": {"name": "xmd", "group": "Ungrouped variables", "definition": "-{b}*{d}+{b}*{c}", "description": "", "templateType": "anything"}, "xmc": {"name": "xmc", "group": "Ungrouped variables", "definition": "-{a}*c+{a}*{d}", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "xmc", "xmd", "f"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "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": "

Select the correct form of the partial fraction:

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["

\\[\\frac{A}{\\simplify{x+{d}}}+\\frac{B}{\\simplify{x+{f}}}+\\frac{C}{\\simplify{(x+{f})^2}}\\]

", "

\\[\\frac{A}{\\simplify{x+{d}}}+\\frac{B}{\\simplify{(x+{d})^2}}+\\frac{C}{\\simplify{x+{f}}}\\]

", "

\\[\\frac{A}{\\simplify{x+{d}}}+\\frac{B}{\\simplify{x+{f}}}+\\frac{C}{\\simplify{x+{f}}}\\]

", "

\\[\\frac{A}{\\simplify{x+{d}}}+\\frac{B}{\\simplify{x+{f}}}\\]

Combine fractions into one:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$A$	$+$	$B$	$+$	$C$	$=$	$A$[[0]]+$B$[[1]]+$C$([[2]])
$\\simplify{x+{d}}$		$\\simplify{x+{f}}$		$\\simplify{(x+{f})^2}$		$\\simplify{(x+{c})*(x+{d})}$

", "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": "(x+{f})^2", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"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": "(x+{d})*(x+{f})", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"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": "x+{d}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "information", "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": "

We now equate the numerators of the original fraction and your fraction from step c), i.e.

\\[\\simplify{{a+b}*x^2+{2*a*f+b*d+b*f+c}*x+{a*f^2+b*d*f+c*d}}=\\simplify{A*(x+{f})^2+B*(x+{d})*(x+{f})+C*(x+{d})}\\]

How can $A$ be calculated most easily? [[0]]

Do it: [[1]] = [[2]]

Therefore $A=$ [[3]]

", "gaps": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["

Substitute $x=\\var{-d}$

", "

Substitute $x=\\var{-f}$

", "

Equate coefficients of $x^2$

", "

Equate coefficients of $x$

", "

Substitute $x=0$

"], "matrix": ["1", 0, 0, 0, 0], "distractors": ["Correct", "No, this would eliminate $A$", "Not helpful because you don't know $B$ yet", "Too many $x$ terms to be useful", "This would involve $A$, $B$ and $C$, so not very useful"]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a*(f-d)^2}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "A*{(f-d)^2}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": [{"name": "a", "value": ""}]}, {"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": "{a}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}, {"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": "

How can $C$ be calculated most easily? [[0]]

Do it: [[1]] = [[2]]

Therefore $C=$ [[3]]

Substitute $x=\\var{-f}$

", "

Substitute $x=\\var{-d}$

", "

Equate coefficients of $x^2$

", "

Equate coefficients of $x$

", "

Substitute $x=0$

"], "matrix": ["1", 0, 0, 0, 0], "distractors": ["Correct", "No, this would eliminate $C$", "Not helpful because $C$ does not involve $x^2$", "Too many $x$ terms to be useful", "This would involve $A$, $B$ and $C$, so not very useful"]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{c*(d-f)}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "C*{d-f}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": [{"name": "c", "value": ""}]}, {"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": "{c}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}, {"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": "

How can $B$ be calculated most easily? [[0]]

Do it: [[1]] = [[2]]

Use your value for $A$ to calculate $B=$ [[3]]

Substitute $x=\\var{-f}$

", "

Substitute $x=\\var{-d}$

", "

Equate coefficients of $x^2$

", "

Equate coefficients of $x$

", "

Substitute $x=0$

"], "matrix": ["0", 0, "1", 0, 0], "distractors": ["No, this would eliminate $B$", "No, this would eliminate $B$", "Correct", "Too many $x$ terms to be useful", "This would involve $A$, $B$ and $C$, so not very useful"]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a+b}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "A+B", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": [{"name": "a", "value": ""}, {"name": "b", "value": ""}]}, {"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": "{b}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}, {"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": "

Therefore

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\simplify{{a+b}x^2+{2af+bd+bf+c}x+{af^2+bdf+cd}}$	$=$	[[0]]	$+$	[[1]]	$+$	[[2]]
$\\simplify{(x+{d})*(x+{f})^2}$		$\\simplify{x+{d}}$		$\\simplify{x+{f}}$		$\\simplify{(x+{f})^2}$

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Resolve the following fraction into partial fractions.

\\[\\frac{\\simplify{{a+b}*x^2+{b*d+c}*x+{a*f+c*d}}}{\\simplify{(x+{d})*(x^2+{f})}}\\]

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "xmd": {"name": "xmd", "group": "Ungrouped variables", "definition": "-{b}*{d}+{b}*{c}", "description": "", "templateType": "anything", "can_override": false}, "xmc": {"name": "xmc", "group": "Ungrouped variables", "definition": "-{a}*c+{a}*{d}", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "xmc", "xmd", "f"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "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": "

Select the correct form of the partial fraction:

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["

\\[\\frac{A}{\\simplify{x+{d}}}+\\frac{Bx+C}{\\simplify{x^2+{f}}}\\]

", "

\\[\\frac{Ax+B}{\\simplify{x+{d}}}+\\frac{C}{\\simplify{x^2+{f}}}\\]

", "

\\[\\frac{A}{\\simplify{x+{d}}}+\\frac{B}{\\simplify{x+{f}}}+\\frac{C}{\\simplify{x-{f}}}\\]

", "

\\[\\frac{A}{\\simplify{x+{d}}}+\\frac{B}{\\simplify{x^2+{f}}}\\]

Combine fractions into one:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$A$	$+$	$Bx+C$	$=$	$A$([[0]])+$(Bx+C)$([[1]])
$\\simplify{x+{d}}$		$\\simplify{x^2+{f}}$		$\\simplify{(x+{d})*(x^2+{f})}$

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We now equate the numerators of the original fraction and your fraction from step b), i.e.

\\[\\simplify{{a+b}*x^2+{b*d+c}*x+{a*f+c*d}}=\\simplify{A*(x^2+{f})+(B*x+C)*(x+{d})}\\]

How can $A$ be calculated most easily? [[0]]

Do it: [[1]]$=\\;$[[2]]

Therefore $A=\\;$[[3]]

Substitute $x=\\var{-d}$

", "

Substitute $x=\\var{-f}$

", "

Equate coefficients of $x^2$

", "

Equate coefficients of $x$

", "

Substitute $x=0$

"], "matrix": ["1", 0, 0, 0, 0], "distractors": ["Correct", "No, this would not eliminate anything", "Not helpful because you don't know $B$ yet", "Not helpful because $A$ is not involved with any $x$ terms", "Not helpful because you don't know $C$ yet"]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a*(d^2+f)}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "A*{d^2+f}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "a", "value": ""}]}, {"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": "{a}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"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": "

How can $B$ be calculated most easily? [[0]]

Do it: [[1]]$=\\;$[[2]]

Use your value of $A$ to calculate $B=\\;$[[3]]

Substitute $x=\\var{-d}$

", "

Substitute $x=\\var{-f}$

", "

Equate coefficients of $x^2$

", "

Equate coefficients of $x$

", "

Substitute $x=0$

"], "matrix": ["0", 0, "1", 0, 0], "distractors": ["No, this would eliminate $B$", "No, this would not eliminate anything", "Correct", "Not helpful because you don't know $C$ yet", "No, this would eliminate $B$"]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a+b}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "A+B", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "a", "value": ""}, {"name": "b", "value": ""}]}, {"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": "{b}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"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": "

How can $C$ be calculated most easily? [[0]]

Do it: [[1]]$=\\;$[[2]]

Use your value of $A$ to calculate $C=\\;$[[3]]

Substitute $x=\\var{-d}$

", "

Substitute $x=\\var{-f}$

", "

Equate coefficients of $x^2$

", "

Substitute $x=0$

"], "matrix": ["0", 0, "0", "1"], "distractors": ["No, this would eliminate $C$", "No, this would not eliminate anything", "No, you have already done this in the previous step", "Correct"]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a*f+c*d}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "A*{f}+C*{d}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "a", "value": ""}, {"name": "c", "value": ""}]}, {"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": "{c}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"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": "

Therefore

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\simplify{{a+b}x^2+{bd+c}x+{af+c*d}}$	$=$	[[0]]	$+$	[[1]]$x\\;+\\;$[[2]]
$\\simplify{(x+{d})*(x^2+{f})}$		$\\simplify{x+{d}}$		$\\simplify{x^2+{f}}$

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This quiz is designed to help you practise the topic of partial fractions. The three questions cover the main types you will encounter:

Linear factors
Repeated linear factor
Quadratic factor

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