// Numbas version: exam_results_page_options
{"name": "Luis's copy of Roots of a cubic real polynomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"variableReplacementStrategy": "originalfirst", "gaps": [{"choices": ["<p>$\\simplify{{a1}+{b1}i}$</p>", "<p>$\\simplify{{x1a1}+{x1b1}i}$</p>", "<p>$\\simplify{{x2a1}+{x2b1}i}$</p>", "<p>$\\simplify{{x3a1}+{x3b1}i}$</p>"], "displayColumns": 4, "distractors": ["\n            \n            \n            <p>Correct!</p>\n            \n            \n            \n            <p>\\[\\begin{eqnarray*}\n            \n            \\simplify{f({z1})}\t&=&\\simplify[std]{{z1}^3+{-2*a1 -c1}*{z1} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z1^3}+{-2*a1 -c1}{z1 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z1^3}+{(( -2) * a1 -c1)*z1^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z1}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\t\n            \n            &=&0\t\t\t\t\t\t\t\t\n            \n            \\end{eqnarray*}\n            \n            \\] Hence is a root.</p>\n            \n            ", "\n            \n            \n            <p>Not the correct choice as :\\[\\begin{eqnarray*}\n            \n            \\simplify{f({z2})}\t&=&\\simplify[std]{{z2}^3+{-2*a1 -c1}*{z2} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z2} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z2^3}+{-2*a1 -c1}{z2 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z2} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z2^3}+{(( -2) * a1 -c1)*z2^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z2}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\t\n            \n            &=&\\simplify[std]{{z2^3+(( -2) * a1 -c1)*z2^2+ (2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z2-c1 * (a1 ^ 2 + b1 ^ 2)}}\\neq 0\t\t\t\t\t\t\t\t\n            \n            \\end{eqnarray*}\\]<br/>Hence not a root</p>\n            \n            ", "\n            \n            \n            <p>Not the correct choice as :\\[\\begin{eqnarray*}\n            \n            \\simplify{f({z3})}\t&=&\\simplify[std]{{z3}^3+{-2*a1 -c1}*{z3} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z3} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z3^3}+{-2*a1 -c1}{z3 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z3} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z3^3}+{(( -2) * a1 -c1)*z3^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z3}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\t\n            \n            &=&\\simplify[std]{{z3^3+(( -2) * a1 -c1)*z3^2+ (2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z3-c1 * (a1 ^ 2 + b1 ^ 2)}}\\neq 0\t\t\t\t\t\t\t\t\n            \n            \\end{eqnarray*}\\]<br/>Hence not a root</p>\n            \n            ", "\n            \n            \n            <p>Not the correct choice as :\\[\\begin{eqnarray*}\n            \n            \\simplify{f({z2})}\t&=&\\simplify[std]{{z4}^3+{-2*a1 -c1}*{z4} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z4} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z4^3}+{-2*a1 -c1}{z4 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z4} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z4^3}+{(( -2) * a1 -c1)*z4^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z4}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\t\n            \n            &=&\\simplify[std]{{z4^3+(( -2) * a1 -c1)*z4^2+ (2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z4-c1 * (a1 ^ 2 + b1 ^ 2)}}\\neq 0\t\t\t\t\t\t\t\t\n            \n            \\end{eqnarray*}\\]<br/>Hence not a root</p>\n            \n            "], "scripts": {}, "maxMarks": 0, "matrix": [1, 0, 0, 0], "variableReplacementStrategy": "originalfirst", "minMarks": 0, "shuffleChoices": true, "variableReplacements": [], "displayType": "radiogroup", "showCorrectAnswer": true, "marks": 0, "type": "1_n_2"}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>Given &nbsp;$\\displaystyle f(z) = \\simplify[std]{z ^ 3 + {( -2) * a1 -c1}*z ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * z -{c1 * (a1 ^ 2 + b1 ^ 2)}}$, one of the following complex numbers is a root $z_1$ of the equation $f(z)=0$.</p>\n        <p>Choose the correct value for $z_1$:[[0]]</p>", "scripts": {}, "marks": 0, "type": "gapfill"}, {"variableReplacementStrategy": "originalfirst", "gaps": [{"answer": "{a1}-{b1}i", "vsetrange": [0, 1], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "checkingaccuracy": 0.001, "answersimplification": "std", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "marks": 1, "checkvariablenames": false, "expectedvariablenames": [], "vsetrangepoints": 5}, {"answer": "{c1}", "vsetrange": [0, 1], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "checkingaccuracy": 0.001, "answersimplification": "std", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "marks": 1, "checkvariablenames": false, "expectedvariablenames": [], "vsetrangepoints": 5}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>The remaining roots of $f(z)$ are:</p>\n        <p>$z_2=\\;\\;$[[0]] (enter the complex root here)</p>\n        <p>$z_3=\\;\\;$[[1]] (enter the real root here)</p>", "scripts": {}, "marks": 0, "type": "gapfill"}], "name": "Luis's copy of Roots of a cubic real polynomial", "ungrouped_variables": ["tx3a1", "f1", "tx3a2", "b2", "tx2a2", "x1b2", "x1b1", "b1", "tx1a1", "tx2a1", "xs2", "xs3", "xs1", "x3a1", "x3a2", "s3", "s2", "s1", "s6", "s5", "s4", "x2a1", "x2a2", "x1a2", "x1a1", "a1", "a2", "z4", "c2", "c1", "z1", "z2", "z3", "x3b1", "x3b2", "x2b1", "x2b2", "tx1a2"], "variables": {"s6": {"description": "", "definition": "random(1,-1)", "templateType": "anything", "name": "s6", "group": "Ungrouped variables"}, "s2": {"description": "", "definition": "random(1,-1)", "templateType": "anything", "name": "s2", "group": "Ungrouped variables"}, "tx3a1": {"description": "", "definition": "s4*random(1..9)", "templateType": "anything", "name": "tx3a1", "group": "Ungrouped variables"}, "z2": {"description": "", "definition": "x1a1+x1b1*i", "templateType": "anything", "name": "z2", "group": "Ungrouped variables"}, "xs1": {"description": "", "definition": "random(1,-1)", "templateType": "anything", "name": "xs1", "group": "Ungrouped variables"}, "z1": {"description": "", "definition": "a1+ b1*i", "templateType": "anything", "name": "z1", "group": "Ungrouped variables"}, "xs2": {"description": "", "definition": "random(1,-1)", "templateType": "anything", "name": "xs2", "group": "Ungrouped variables"}, "x2a1": {"description": "", "definition": "if(tx2a1=a1,a1+1,tx2a1)", "templateType": "anything", "name": "x2a1", "group": "Ungrouped variables"}, "x1b1": {"description": "", "definition": "xs1*random(1..9)", "templateType": "anything", "name": "x1b1", "group": "Ungrouped variables"}, "s5": {"description": "", "definition": "random(1,-1)", "templateType": "anything", "name": "s5", "group": "Ungrouped variables"}, "b1": {"description": "", "definition": "if(abs(a1)=f1,f1+1,f1)", "templateType": "anything", "name": "b1", "group": "Ungrouped variables"}, "x1b2": {"description": "", "definition": "xs2*random(1..9)", "templateType": "anything", "name": "x1b2", "group": "Ungrouped variables"}, "tx1a1": {"description": "", "definition": "s1*random(1..9)", "templateType": "anything", "name": "tx1a1", "group": "Ungrouped variables"}, "x3b1": {"description": "", "definition": "xs3*random(1..9)", "templateType": "anything", "name": "x3b1", "group": "Ungrouped variables"}, "x1a1": {"description": "", "definition": "if(tx1a1=a1,a1+1,tx1a1)", "templateType": "anything", "name": "x1a1", "group": "Ungrouped variables"}, "a2": {"description": "", "definition": "s4*random(1..9)", "templateType": "anything", "name": "a2", "group": "Ungrouped variables"}, "s4": {"description": "", "definition": "random(1,-1)", "templateType": "anything", "name": "s4", "group": "Ungrouped variables"}, "x3a1": {"description": "", "definition": "if(tx3a1=a1,a1+1,tx3a1)", "templateType": "anything", "name": "x3a1", "group": "Ungrouped variables"}, "x3b2": {"description": "", "definition": "xs1*random(1..9)", "templateType": "anything", "name": "x3b2", "group": "Ungrouped variables"}, "x2b2": {"description": "", "definition": "xs3*random(1..9)", "templateType": "anything", "name": "x2b2", "group": "Ungrouped variables"}, "z4": {"description": "", "definition": "x3a1+x3b1*i", "templateType": "anything", "name": "z4", "group": "Ungrouped variables"}, "tx3a2": {"description": "", "definition": "s6*random(1..9)", "templateType": "anything", "name": "tx3a2", "group": "Ungrouped variables"}, "s1": {"description": "", "definition": "random(1,-1)", "templateType": "anything", "name": "s1", "group": "Ungrouped variables"}, "b2": {"description": "", "definition": "s5*random(1..9)", "templateType": "anything", "name": "b2", "group": "Ungrouped variables"}, "x2a2": {"description": "", "definition": "if(tx2a2=a2,a2+1,tx2a2)", "templateType": "anything", "name": "x2a2", "group": "Ungrouped variables"}, "f1": {"description": "", "definition": "random(1..8)", "templateType": "anything", "name": "f1", "group": "Ungrouped variables"}, "x2b1": {"description": "", "definition": "xs2*random(1..9)", "templateType": "anything", "name": "x2b1", "group": "Ungrouped variables"}, "c1": {"description": "", "definition": "s3*random(1..9)", "templateType": "anything", "name": "c1", "group": "Ungrouped variables"}, "c2": {"description": "", "definition": "s6*random(1..9)", "templateType": "anything", "name": "c2", "group": "Ungrouped variables"}, "tx2a1": {"description": "", "definition": "s3*random(1..9)", "templateType": "anything", "name": "tx2a1", "group": "Ungrouped variables"}, "a1": {"description": "", "definition": "s1*random(1..9)", "templateType": "anything", "name": "a1", "group": "Ungrouped variables"}, "s3": {"description": "", "definition": "random(1,-1)", "templateType": "anything", "name": "s3", "group": "Ungrouped variables"}, "x1a2": {"description": "", "definition": "if(tx1a2=a2,a2+1,tx1a2)", "templateType": "anything", "name": "x1a2", "group": "Ungrouped variables"}, "tx1a2": {"description": "", "definition": "s3*random(1..9)", "templateType": "anything", "name": "tx1a2", "group": "Ungrouped variables"}, "x3a2": {"description": "", "definition": "if(tx3a2=a2,a2+1,tx3a2)", "templateType": "anything", "name": "x3a2", "group": "Ungrouped variables"}, "xs3": {"description": "", "definition": "random(1,-1)", "templateType": "anything", "name": "xs3", "group": "Ungrouped variables"}, "z3": {"description": "", "definition": "x2a1+x2b1*i", "templateType": "anything", "name": "z3", "group": "Ungrouped variables"}, "tx2a2": {"description": "", "definition": "s4*random(1..9)", "templateType": "anything", "name": "tx2a2", "group": "Ungrouped variables"}}, "preamble": {"css": "", "js": ""}, "statement": "<p>From the list of complex numbers, choose the one which is a root of the given equation $f(z)=0$ , and hence find all roots.</p>", "metadata": {"description": "<p>Using a given list of four complex numbers, find by inspection the one that is a root of a given cubic real polynomial and hence find the other roots.</p>", "notes": "<p><strong>15/07/2015:</strong></p>\n<p>Added tags.</p>\n<p><strong>5/07/2012:</strong></p>\n<p>Added tags.</p>\n<p>Question appears to work correctly.</p>\n<p>Changed grammar in Advice.</p>\n<p><strong>9/07/2012:</strong></p>\n<p>Improved Advice display.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "variable_groups": [], "question_groups": [{"questions": [], "pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered"}], "showQuestionGroupNames": false, "advice": "<p><strong>a) Finding a root</strong>.</p>\n    <p>In order to find which one of the four choices is a root you have to evaluate $f(z)$ for each choice.If you find for a choice of $z$ that $f(z)=0$ then that choice of $z$ is a root of the equation.</p>\n    <p>Note that\\[\\begin{eqnarray*} \\simplify{f({z1})} &amp;=&amp;\\simplify[std]{{z1}^3+{-2*a1 -c1}*{z1} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\ &amp;=&amp;\\simplify[std]{{z1^3}+{-2*a1 -c1}{z1 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\ &amp;=&amp;\\simplify[std]{{z1^3}+{(( -2) * a1 -c1)*z1^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z1}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\ &amp;=&amp;0 \\end{eqnarray*}. \\]So of the list of choices $z_1=\\var{z1}$ is a root.</p>\n    <p><strong>b) The other roots</strong></p>\n    <p>Now that you have found a complex root it is very simple to find another complex root.</p>\n    <p>Since $f(z)$ is a polynomial with real coefficients then if $z=z_0$ is a root we have that the conjugate $z=\\overline{z_0}$ is also a root.</p>\n    <p>Hence the complex number $z_2=\\overline{\\var{z1}}=\\var{conj(z1)}$ is a root.</p>\n    <p>To find the real root $z_3=c$ we note that the constant term of\\[f(z) =(z-z_1)(z-z_2)(z-c)\\]is $-z_1z_2c = -(\\var{z1})(\\var{conj(z1)})c=\\var{-z1*conj(z1)}c$.</p>\n    <p>But we know that the constant term of $f(z)$ is $\\simplify{-{c1 * (a1 ^ 2 + b1 ^ 2)}} $.</p>\n    <p>Hence \\[\\begin{eqnarray*} \\var{-z1*conj(z1)}c &amp;=&amp;\\simplify{-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\ \\Rightarrow c &amp;=&amp;&nbsp; \\simplify[]{{c1 * (a1 ^ 2 + b1 ^ 2)}/{abs(z1^2)}}\\\\ &amp;=&amp;\\var{c1} \\end{eqnarray*} \\]</p>", "type": "question", "tags": ["checked2015", "complex numbers", "complex roots of real  polynomials", "conjugate roots", "mas1602", "MAS1602", "roots of polynomial equations", "roots of polynomials", "roots of real polynomials"], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}