// Numbas version: exam_results_page_options
{"name": "Divide Polynomials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["be", "s2", "s1", "m", "al", "n", "s", "r", "t"], "functions": {}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "<p>Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n  <p>Divide $\\displaystyle{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}$ by $\\simplify[std]{{r}x+{s}}$ so that:<br/>\\[\\frac{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}{\\simplify[std]{{r}x+{s}}}=q(x)+\\frac{r}{\\simplify[std]{{r}x+{s}}}\\]</p>\n  <p>where $q(x)$ is the quotient polynomial and $r$ is the remainder ($r$ is a constant).</p>\n  <p>The coefficients of $q(x)$ are integers, do not input as decimals.</p>\n  ", "variables": {"s2": {"templateType": "anything", "description": "", "definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s2"}, "t": {"templateType": "anything", "description": "", "definition": "s2*random(-9..9)", "group": "Ungrouped variables", "name": "t"}, "m": {"templateType": "anything", "description": "", "definition": "1", "group": "Ungrouped variables", "name": "m"}, "r": {"templateType": "anything", "description": "", "definition": "1", "group": "Ungrouped variables", "name": "r"}, "n": {"templateType": "anything", "description": "", "definition": "random(-9..9)", "group": "Ungrouped variables", "name": "n"}, "al": {"templateType": "anything", "description": "", "definition": "random(-9..9)", "group": "Ungrouped variables", "name": "al"}, "s1": {"templateType": "anything", "description": "", "definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s1"}, "s": {"templateType": "anything", "description": "", "definition": "s1*random(1..9)", "group": "Ungrouped variables", "name": "s"}, "be": {"templateType": "anything", "description": "", "definition": "random(-9..9)", "group": "Ungrouped variables", "name": "be"}}, "tags": [], "parts": [{"customName": "", "customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "prompt": "\n      \n      \n      <p>$q(x)=\\;\\;$[[0]]</p>\n      \n      \n      \n      <p>Input all numbers as integers  and not as decimals.</p>\n      \n      \n      \n      <p>$r=\\;\\;$[[1]]</p>\n      \n      \n      ", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "unitTests": [], "useCustomName": false, "gaps": [{"customName": "", "customMarkingAlgorithm": "", "failureRate": 1, "checkingType": "absdiff", "showCorrectAnswer": true, "checkVariableNames": false, "vsetRange": [0, 1], "showFeedbackIcon": true, "vsetRangePoints": 5, "notallowed": {"message": "<p>Input numbers as integers not decimals.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "checkingAccuracy": 0.001, "scripts": {}, "answer": "(({m} * (x ^ 2)) + ({n} * x) + {t})", "valuegenerators": [{"value": "", "name": "x"}], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "answerSimplification": "std", "marks": 1, "unitTests": [], "showPreview": true, "useCustomName": false, "variableReplacements": []}, {"customName": "", "customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{t*n+be-t*s}", "allowFractions": false, "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "showFractionHint": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 2, "unitTests": [], "minValue": "{t*n+be-t*s}", "useCustomName": false, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}], "variableReplacements": []}], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "advice": "<p>We have:</p>\n<p>\\[\\begin{eqnarray*} \\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}&amp;=&amp;\\simplify[std]{(x+{s})x^2+{n}x^2+{n*s+t}x+{t*n+be}}\\\\&amp;=&amp;\\simplify[std]{(x+{s})x^2+(x+{s})*{n}x+{t}x+{t*n+be}}\\\\ &amp;=&amp;\\simplify[std]{(x+{s})x^2+(x+{s})*{n}x+(x+{s})*{t}+{t*n+be-s*t}}\\\\ &amp;=&amp;\\simplify[std]{(x+{s})(x^2+{n}x+{t})+{t*n+be-s*t}} \\end{eqnarray*} \\]</p>\n<p>Hence<br/>\\[\\frac{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}{\\simplify[std]{{r}x+{s}}}=\\simplify[std]{x^2+{n}x+{t}+{t*n+be-s*t}/({r}x+{s})}\\]</p>", "name": "Divide Polynomials", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}