// Numbas version: exam_results_page_options
{"name": "Factorise rational polynomials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "r"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,1,-1])", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "c"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "d"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "a1", "r"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Factorise rational polynomials", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "X+{a}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{r}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>$\\simplify{f(X) = X ^ 5 + {1 + a} * X ^ 4 + {1 + a} * X ^ 3 + {1 + a} * X ^ 2 + {1 + a} * X + {a + r}}$</p>\n<p>$\\simplify{g(X) = X ^ 4 + X ^ 3 + X ^ 2 + X + 1}$</p>\n<p>$q(X)=$ [[0]]</p>\n<p>$r(X)=$ [[1]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "X-{a}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{-r}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>$\\simplify{f(X) = X ^ 5 + { -1 -a} * X ^ 4 + {1 + a} * X ^ 3 + { -1 -a} * X ^ 2 + {1 + a} * X + { -a -r}}$</p>\n<p>$\\simplify{g(X) = X ^ 4 - X ^ 3 + X ^ 2 - X + 1}$</p>\n<p>$q(X)=$ [[0]]</p>\n<p>$r(X)=$ [[1]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "X^3/{a1}+{d}/{a1}*X-{b}/{a1^2}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>Enter numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}, {"answer": "-{b*d}/{a1}*X+{b^2+a1^2*c}/{a1^2}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>Enter numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>$\\simplify{f(X) = X ^ 6 + {d} * X ^ 4 + {c}}$</p>\n<p>$\\simplify{g(X) = {a1} * X ^ 3 + {b}}$</p>\n<p>Remember to enter all numbers as rationals i.e. as fractions or integers and not as decimals.</p>\n<p>$q(X)=$ [[0]]</p>\n<p>$r(X)=$ [[1]]</p>\n<p></p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Let $\\mathbb{Q}$ be the rational numbers. This exercise considers polynomials over $\\mathbb{Q}$.</p>\n<p>In each of the following examples, find polynomials $q(X)$ and $r(X)$ over $\\mathbb{Q}$ such that $f(X)=q(X)g(X)+r(X)$ and &nbsp; $\\operatorname{deg}r(X) \\lt \\operatorname{deg}g(X)$.</p>\n<p>Enter all numbers as rationals, i.e. as fractions or integers and not as decimals.</p>\n<p></p>", "tags": ["checked2015", "MAS2213"], "rulesets": {"std": ["all", "!noleadingminus", "fractionnumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given polynomial $f(X)$, $g(X)$ over $\\mathbb{Q}$, find polynomials $q(X)$ and $r(X)$ over $\\mathbb{Q}$ such that $f(X)=q(X)g(X)+r(X)$ and $\\operatorname{deg}r(X) \\lt \\operatorname{deg}g(X)$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h4>a)</h4>\n<p>We have:</p>\n<p>\\[ \\simplify{f(X) = X ^ 5 + {1 + a} * X ^ 4 + {1 + a} * X ^ 3 + {1 + a} * X ^ 2 + {1 + a} * X + {a + r} = (X + {a}) * (X ^ 4 + X ^ 3 + X ^ 2 + X + 1) + {r}} \\]</p>\n<p>Hence $\\simplify{g(X)=X+{a}}$, $r=\\var{r}$.</p>\n<h4>b)</h4>\n<p>Similarly:</p>\n<p>\\[ \\simplify{f(X) = X ^ 5 + { -1 -a} * X ^ 4 + {1 + a} * X ^ 3 + { -1 -a} * X ^ 2 + {1 + a} * X + { -a -r} = (X + { -a}) * (X ^ 4 -(X ^ 3) + X ^ 2 -X + 1) + { -r}} \\]</p>\n<p>Hence $\\simplify{g(X)=X-{a}}$, $r=\\var{-r}$.</p>\n<h4>c)</h4>\n<p>\\begin{align}<br/>f(X) &amp;= \\simplify{X ^ 6 + {d} * X ^ 4 + {c}} \\\\<br/>&amp;= \\simplify[std]{({a1} * X ^ 3 + {b}) * (X ^ 3 / {a1} + &nbsp;({d} / {a1}) * X &nbsp;-({b} / {a1 ^ 2})) + {b^2 + c * a1 ^ 2} / {a1 ^ 2} - &nbsp;({b * d} / {a1}) * X}<br/>\\end{align}</p>\n<p>Hence</p>\n<p>\\begin{align} \\displaystyle g(X) &amp;=&nbsp;\\simplify[std]{X ^ 3 / {a1} + &nbsp;({d} / {a1}) * X &nbsp;-({b} / {a1 ^ 2})}, &amp; r(X) &amp;= \\simplify[std]{{b^2 + c * a1 ^ 2} / {a1 ^ 2} - &nbsp;({b * d} / {a1}) * X} \\end{align}</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}