Rewrite the expression $\\frac{n}{(x+a)(x+b)^2}$ as partial fractions in the form $\\frac{A}{x+a}+\\frac{B}{x+b}+\\frac{C}{(x+b)^2}$.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Rewrite the following expression as partial fractions:
\n\\[ \\simplify{{n}/((x+{a})(x+{b})^2)}. \\]
", "advice": "To express \\[ \\simplify{{n}/((x+{a})(x+{b})^2)} \\] as partial fractions, we want to set this equal to the sum of three fractions with denominators $\\simplify{x+{a}}$, $\\simplify{x+{b}}$, and $\\simplify{(x+{b})^2}$. Since we have a distinct linear factor and a repeated linear factor, this tells us that the numerators will be constants, which we will call $A$, $B$, and $C$:
\n\\[ \\simplify{{n}/((x+{a})(x+{b})^2)} = \\simplify{A/(x+{a}) + B/(x+{b})+ C/(x+{b})^2}.\\]
\nTo find the values of $A$, $B$, and $C$, we want to first multiply this equation by the denominator of the left-hand side. This gives
\n\\[ \\simplify{{n}=A(x+{b})^2+B(x+{a})(x+{b}) + C(x+{a})}.\\]
\n(Note: To find $A$, $B$, and $C$, we will use a combination of choosing suitable values of $x$ to eliminate terms, and equating coefficients. It can be solved by only equating coefficients, but this is a more efficient process.)
\n\nTo find $A$, we can eliminate $B$ and $C$ by setting $x=\\var{-a}$:
\n\\[ \\simplify{{n}=A{(b-a)^2}} \\implies A=\\simplify[fractionNumbers]{{Asol}}.\\]
\nTo find $C$, we can eliminate $A$ and $B$ by setting $x=\\var{-b}$:
\n\\[ \\simplify{{n}=C{(a-b)}} \\implies C=\\simplify[fractionNumbers]{{Csol}}.\\]
\nFinally, by equating coefficients of the $x^2$-terms we can find $B$:
\n\\[ (x^2): \\quad 0 = \\simplify{A+B} \\implies B=-A. \\]
\nTherefore, \\[ B=\\simplify[fractionNumbers]{{Bsol}}, \\]
\nand
\n{check}
\n", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-6..6)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-6..6 except a)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "Asol": {"name": "Asol", "group": "Ungrouped variables", "definition": "n/((b-a)(b-a))", "description": "", "templateType": "anything", "can_override": false}, "Bsol": {"name": "Bsol", "group": "Ungrouped variables", "definition": "-Asol", "description": "", "templateType": "anything", "can_override": false}, "Csol": {"name": "Csol", "group": "Ungrouped variables", "definition": "n/(a-b)", "description": "", "templateType": "anything", "can_override": false}, "check": {"name": "check", "group": "Ungrouped variables", "definition": "if(Asol=round(Asol) and Bsol=round(Bsol),'{sol1}','{sol2}')", "description": "", "templateType": "anything", "can_override": false}, "sol1": {"name": "sol1", "group": "Ungrouped variables", "definition": "\"
\\\\[ \\\\simplify{{n}/((x+{a})(x+{b})^2)} = \\\\simplify{{Asol}/(x+{a})+{Bsol}/(x+{b})+{Csol}/(x+{b})^2}.\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "sol2": {"name": "sol2", "group": "Ungrouped variables", "definition": "\"\\\\[ \\\\simplify{{n}/((x+{a})(x+{b})^2)} = \\\\simplify[all,fractionNumbers]{{n}/({(b-a)^2}(x+{a}))-{n}/({(b-a)^2}(x+{b}))+{n}/({(a-b)}(x+{b})^2)}.\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "n", "Asol", "Bsol", "Csol", "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[[0]]
", "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": "{n}/({(b-a)^2}(x+{a}))-{n}/({(b-a)^2}(x+{b}))+{n}/({(a-b)}(x+{b})^2)", "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`?*($n`?x+`+-$n`?)^2) `| `+-$n/`+-$n*(`+-$n`?/($n`?*($n`?x+`+-$n`?))+`+-$n`?/($n`?*($n`?x+`+-$n`?))+`+-$n`?/($n`?*($n`?x+`+-$n`?)^2))", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "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/"}]}