// Numbas version: exam_results_page_options
{"name": "Find greatest common divisor of two polynomials over the rationals", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "r1"}, "d0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "r2*a2*a3+n2", "description": "", "name": "d0"}, "n1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "n*r1", "description": "", "name": "n1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "a1"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a2*r2+a3*r2+a2*a3+n1", "description": "", "name": "d1"}, 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"definition": "1+a1*(r2+a2+a3)+n1+r2*(a2+a3)+a2*a3", "description": "", "name": "c2"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "a2"}, "c0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1*r2*a2*a3+a1*n2+r2*a3", "description": "", "name": "c0"}}, "ungrouped_variables": ["r1", "r2", "n", "a1", "a3", "a2", "n1", "c3", "c2", "n2", "c0", "d2", "c1", "d0", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Find greatest common divisor of two polynomials over the rationals", "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "X+{r2}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>Consider $f(X)$ and $g(X)$ as polynomials over the rational numbers $\\mathbb{Q}$.</p>\n<p>Find their greatest common divisor (GCD).</p>\n<p>GCD=[[0]].</p>\n<p>You must input your answer as a &nbsp;monic polynomial over $\\mathbb{Q}$ .&nbsp;</p>\n<p></p>\n<p></p>\n<p></p>\n<p>Click on Steps to obtain information on monic polynomials.</p>", "steps": [{"type": "information", "prompt": "<p>Normalized Polynomials.</p>\n<p></p>\n<p>A monic polynomial over $\\mathbb{Q}$ &nbsp;has the coefficient of the greatest power equalling 1.</p>\n<p>If a polynomial over $\\mathbb{Q}$ has the coefficient of the greatest power $a \\neq 1$, then you multiply the polynomial by $\\frac{1}{a}$.</p>\n<p></p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Let \\[\\begin{align}\\\\<br/>f(X)&amp;=\\simplify{ X ^ 4 + {c3} * X ^ 3 + {c2} * X ^ 2 + {c1} * X + {c0}}\\\\\\\\<br/>g(X)&amp;=\\simplify{X ^ 3 + {d2} * X ^ 2 + {d1} * X + {d0}}\\end{align}\\]</p>", "tags": ["checked2015", "factorize polynomials", "fields", "gcd of polynomials", "MAS2213", "normalized polyno9mials", "polynomials over fields", "polynomials over the rationals"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>$f(X)$ and $g(X)$ as polynomials over the rational numbers $\\mathbb{Q}$.</p>\n<p>Find their greatest common divisor (GCD) and enter as a normalized polynomial.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>First we divide $f(X)$ by $g(X)$ to get the remainder $r_1(X) =\\simplify{ X ^ 2 + {r2 + a3} * X + {r2 * a3}}$ :</p>\n<p>\\[f(X) = \\simplify{X ^ 4 + {c3} * X ^ 3 + {c2} * X ^ 2 + {c1} * X + {c0} = (X + {a1}) * (X ^ 3 + {d2} * X ^ 2 + {d1} * X + {d0}) + X ^ 2 + {r2 + a3} * X + {r2 * a3}}\\]</p>\n<p>Next we divide $g(X)$ by $r_1(X)$ to get the remainder $ r_2(X) =\\simplify{ {n1} * X + {n2}}$ :</p>\n<p>\\[g(X) = \\simplify{X ^ 3 + {d2} * X ^ 2 + {d1} * X + {d0} = (X + {a2}) * (X ^ 2 + {r2 + a3} * X + {r2 * a3}) + {n1} * X + {n2}}\\]</p>\n<p>Now divide $r_1(X)$ by $r_2(X)$ to obtain:</p>\n<p>\\[r _1(X) =\\simplify{ X ^ 2 + {r2 + a3} * X + {r2 * a3} = (1 / {n}) * (X + {a3}) * ({n1} * X + {n2})}\\]</p>\n<p>As we have zero remainder, the last non-zero remainder is the GCD and this is $\\simplify{{n1} * X + {n2}}$.</p>\n<p>But we have to make this into a monic polynomial and the required monic GCD $\\simplify{X+{r2}}$ is given by multiplying by $\\frac{1}{\\var{n1}}$ so that the coefficient of the highest degree term is 1.</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/"}]}