// Numbas version: finer_feedback_settings
{"name": "Partial Fractions - quadratic term in denominator", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["coeff0", "coeff1", "a", "a1", "a2", "a3", "bcoeff0", "bcoeff1", "b", "b1", "b2", "b3", "ccoeff0", "ccoeff1", "c", "c1", "c2", "c3"], "name": "Partial Fractions - quadratic term in denominator", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>Suppose we want to express&nbsp;</p>\n<p>$\\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})/((x^2+{coeff1}x+{coeff0})(x-{a}))}$</p>\n<p>as the sum of its partial fractions.</p>\n<p>When 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&nbsp;</p>\n<p>$\\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})/((x^2+{coeff1}x+{coeff0})(x-{a}))}=\\simplify{(Ax+B)/(x^2+{coeff1}x+{coeff0})+c/(x+{a})}$</p>\n<p>We multiply both sides by $\\simplify{((x^2+{coeff1}x+{coeff0})(x-{a}))}$ to give&nbsp;$\\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})}=\\simplify{(Ax+B)(x+{a})+C(x^2+{coeff1}x+{coeff0})}$</p>\n<p>By evaluating both sides at $\\simplify{x=-{a}}$ we can find C$=\\var{a3}$</p>\n<p>Then by comparing coefficients we find A$=\\var{a1}$ and B$=\\var{a2}$</p>\n<p></p>\n<p>Apply the same method for b) and c)</p>", "rulesets": {}, "parts": [{"prompt": "<p>Express the following as a sum of partial fractions</p>\n<p>$\\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})/((x^2+{coeff1}x+{coeff0})(x-{a}))}$</p>\n<p>[[0]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Please split into partial fractions</p>", "showStrings": false, "strings": ["{a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a}"], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "(({a1}x+{a2})/(x^2+{coeff1}x+{coeff0}))+({a3}/(x-{a}))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Express the following as a sum of partial fractions</p>\n<p>$\\simplify{({b1+b3}x^2+{b2-b*b1+b3*bcoeff1}x+{b3*bcoeff0-b2*b})/((x^2+{bcoeff1}x+{bcoeff0})(x-{b}))}$</p>\n<p>[[0]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "(({b1}x+{b2})/(x^2+{bcoeff1}x+{bcoeff0}))+({b3}/(x-{b}))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Express the following as a sum of partial fractions</p>\n<p>$\\simplify{({c1+c3}x^2+{c2-c*c1+c3*ccoeff1}x+{c3*ccoeff0-c2*c})/((x^2+{ccoeff1}x+{ccoeff0})(x-{c}))}$</p>\n<p>[[0]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "(({c1}x+{c2})/(x^2+{ccoeff1}x+{ccoeff0}))+({c3}/(x-{c}))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": "10000", "condition": "coeff1^2-4coeff0<0\nand\nbcoeff1^2-4bcoeff0<0\nand\nccoeff1^2-4ccoeff0<0\nand\nb2-b*b1+b3*bcoeff1=0\n"}, "variables": {"a": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "bcoeff0": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "bcoeff0", "description": ""}, "c": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "bcoeff1": 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"random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "b3": {"definition": "-b1", "templateType": "anything", "group": "Ungrouped variables", "name": "b3", "description": ""}, "c3": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c3", "description": ""}, "c2": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "description": ""}, "coeff1": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "coeff1", "description": ""}, "coeff0": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "coeff0", "description": ""}, "c1": {"definition": "-c3", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}