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{"name": "JD's copy of Finding the full factorisation of a polynomial, using the Factor Theorem and long division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "metadata": {"description": "<p>Use a given factor of a polynomial to find the full factorisation of the polynomial through long division.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "tags": [], "functions": {}, "variables": {"y": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-2..3 except 0)", "name": "y", "description": "<p>Factor 3.</p>"}, "u": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-2..3 except 0 except -y)", "name": "u", "description": "<p>Factor 2.</p>"}, "z": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-3..3 except 0)", "name": "z", "description": "<p>Factor 1.</p>"}}, "statement": "<p>The factor theorem states that if $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.&nbsp;</p>", "advice": "<p>For this question, we are given that $(\\simplify{x+{z}})$ is a factor of the polynomial</p>\n<p>\\[p(x) = \\simplify{x^3+({y}+{u}+{z})*x^2+({y}{u}+{z}{u}+{y}{z})*x+{y}{u}{z}},\\]</p>\n<p>and we are then asked to find the full factorisation of $p(x)$.</p>\n<p>We know that $(\\simplify{x+{z}})$ is a factor of $p(x)$, so we can calculate the other factors of $p(x)$ through long division.</p>\n<p>\\[<br/>\\begin{align}<br/>&amp;\\simplify{x^2+({u}+{y})x+{u}{y}}\\\\<br/>\\simplify{x+{z}} \\; &amp;\\overline{)\\simplify{x^3+({y}+{u}+{z})*x^2+({y}{u}+{z}{u}+{y}{z})*x+{y}{u}{z}}}\\\\<br/>&amp;\\;\\,<br/>\\simplify{x^3+{z}x^2}\\\\<br/>&amp;\\qquad\\quad<br/>\\overline{\\simplify[all,noLeadingMinus]{({u}+{y})x^2+({y}{u}+{z}{u}+{y}{z})*x+{y}{u}{z}}}\\\\<br/>&amp;\\qquad\\quad<br/>\\simplify[all,noLeadingMinus]{({u}+{y})x^2+({u}{z}+{z}{y})x}\\\\<br/>&amp;\\qquad\\quad\\quad\\quad\\quad<br/>\\overline{\\simplify[all,noLeadingMinus]{{y}{u}x+{y}{u}{z}}}\\\\<br/>&amp;\\qquad\\quad\\quad\\quad\\quad<br/>\\simplify[all,noLeadingMinus]{{y}{u}x+{y}{u}{z}}\\\\<br/>&amp;\\qquad\\qquad\\quad\\quad\\quad<br/>\\overline{0.}<br/>\\end{align}<br/>\\]</p>\n<p>We can then factorise $\\simplify{x^2+({u}+{y})x+{u}{y}}$ into</p>\n<p>\\[\\simplify{x^2+({u}+{y})x+{u}{y}} =(\\simplify{x+{y}})(\\simplify{x+{u}}).\\]</p>\n<p>Therefore, the full factorisation&nbsp;of $p(x)$ is</p>\n<p>\\[<br/>\\begin{align}<br/>p(x) &amp;= \\simplify{x^3+({y}+{u}+{z})*x^2+({y}{u}+{z}{u}+{y}{z})*x+{y}{u}{z}},\\\\<br/>&amp;= (\\simplify{x+{y}})(\\simplify{x+{z}})(\\simplify{x+{u}}).<br/>\\end{align}<br/>\\]</p>", "variable_groups": [], "rulesets": {}, "name": "JD's copy of Finding the full factorisation of a polynomial, using the Factor Theorem and long division", "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["y", "u", "z"], "parts": [{"answer": "(x+{y})(x+{u})(x+{z})", "useCustomName": false, "customMarkingAlgorithm": "", "customName": "", "vsetRange": [0, 1], "showPreview": true, "vsetRangePoints": 5, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "unitTests": [], "valuegenerators": [{"value": "", "name": "x"}], "scripts": {}, "marks": "2", "showFeedbackIcon": true, "mustmatchpattern": {"pattern": "((`+-x^$n`? + `+- $n)^$n`?)`* * $z", "partialCredit": 0, "message": "Your answer is not fully factorised.", "nameToCompare": ""}, "type": "jme", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "prompt": "<p>Given that $(\\simplify{x+{z}})$ is a factor of \\[p(x) = \\simplify{x^3+({y}+{u}+{z})*x^2+({y}*{u}+{z}*{u}+{y}*{z})*x+{y}*{u}*{z}}.\\]</p>\n<p>Find the full factorisation of $p(x)$.</p>", "checkVariableNames": false}], "variablesTest": {"condition": "", "maxRuns": "192"}, "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "JD Ichwan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3389/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "JD Ichwan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3389/"}]}