// Numbas version: exam_results_page_options
{"name": "Dividing Polynomials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Dividing Polynomials", "tags": ["algebra", "algebraic manipulation", "dividing polynomials", "division of polynomials", "polynomial division", "quotient polynomial", "remainder polynomial"], "advice": "\n    <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>\n    \n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"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        \n      ", "gaps": [{"notallowed": {"message": "<p>Input numbers as integers not decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "(({m} * (x ^ 2)) + ({n} * x) + {t})", "type": "jme"}, {"minvalue": "{t*n+be-t*s}", "type": "numberentry", "maxvalue": "{t*n+be-t*s}", "marks": 2.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "extensions": [], "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    \n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"be": {"definition": "random(-9..9)", "name": "be"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "m": {"definition": 1.0, "name": "m"}, "al": {"definition": "random(-9..9)", "name": "al"}, "n": {"definition": "random(-9..9)", "name": "n"}, "s": {"definition": "s1*random(1..9)", "name": "s"}, "r": {"definition": 1.0, "name": "r"}, "t": {"definition": "s2*random(-9..9)", "name": "t"}}, "metadata": {"notes": "\n            \t\t            \t\t<p><strong>28/6/2012:</strong></p>\n            \t\t            \t\t    <p>Changed the divisor to $x+a$ where $a \\neq 0$, before this $a=0$ was allowed making the question easy.</p>\n            \t\t            \t\t    <p>Added decimal point . as forbidden string to stop decimal input (is this necessary?)</p>\n            \t\t            \t\t    <p>Added tags.</p>\n            \t\t            \t\t    <p>The solution is given in terms of writing the dividend polynomial as powers of the linear divisor polynomial rather than using standard polynomial long division.</p>\n            \t\t            \t\t    <p><strong>18/07/2012:</strong></p>\n            \t\t            \t\t    <p>Added description.</p>\n            \t\t            \t\t    <p><strong>23/07/2012:</strong></p>\n            \t\t            \t\t    <p>Added tags.</p>\n            \t\t            \t\t    <p>Question appears to be working correctly.</p>\n            \t\t            \t\t    <p>&nbsp;</p>\n            \t\t            \n            \t\t", "description": "<p>Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}