// Numbas version: exam_results_page_options
{"name": "The Fundamental Theorem of Algebra", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["degree", "c", "poly"], "name": "The Fundamental Theorem of Algebra", "tags": ["algebra", "Algebra", "Fundamental", "Polynomial", "polynomial", "roots"], "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "1", "prompt": "<p>The polynomial {poly} has&nbsp;[[0]]&nbsp;(complex) roots (counted with multiplicity).</p>\n<p>That is, it can be written as&nbsp;the&nbsp;product of a constant (complex) factor and&nbsp;[[0]]&nbsp;linear factors $x-r_i$ where $r_i\\in \\mathbb{C}$ is a root of the polynomial.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{degree}", "minValue": "{degree}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "<p>The Fundamental Theorem of Algebra can be&nbsp;stated as follows: every non-zero, single-variable,&nbsp;degree $n$&nbsp;polynomial with complex coefficients has, counted with multiplicity, exactly&nbsp;$n$&nbsp;roots.&nbsp;</p>\n<p></p>\n<p>Since real numbers are complex numbers (just with an imaginary part of zero) and the highest power in the given polynomial&nbsp;is $\\var{degree}$, the Fundamental Theorem of Algebra can be applied&nbsp;and therefore the polynomial has exactly&nbsp;$\\var{degree}$ roots (counted with multiplicity). That is, there isn't necessarily&nbsp;$\\var{degree}$ unique roots but the polynomial can be written as the product of $\\var{degree}$ linear factors&nbsp;$x-r_i$ where $r_i\\in \\mathbb{C}$ is a root of the polynomial.</p>\n<p></p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"c": {"definition": "repeat(random(0,random(-12..12 except 0)),degree)+[random(-12..12 except 0)]", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "degree": {"definition": "random(2..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "degree", "description": ""}, "poly": {"definition": "if(degree=2,'\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2}\\$',\nif(degree=3, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3}\\$',\nif(degree=4, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4}\\$',\nif(degree=5, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5}\\$',\nif(degree=6, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6}\\$',\nif(degree=7, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7}\\$',\nif(degree=8, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8}\\$',\nif(degree=9, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8+{c[9]}x^9}\\$',\n'\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8+{c[9]}x^9+{c[10]}x^{10}}\\$'))))))))                ", "templateType": "anything", "group": "Ungrouped variables", "name": "poly", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}