// Numbas version: exam_results_page_options
{"name": "Factor Theorem and polynomials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "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>Using the factor theorem, we know that if $(x-a)$ is&nbsp;a factor of a polynomial $f(x)$, then $f(a)=0$.</p>\n<p>We are given that $(\\simplify{x+{d}})$ is a factor of $g(x) =&nbsp;\\simplify{{w}*x^3+({w}{d}+{a}+{w}{b})*x^2+({a}{d}+{w}{b}{d}+{a}{b})*x+m}$.</p>\n<p>By the factor theorem, this means that $g(\\simplify{-{d}}) = 0$.</p>\n<p>Substituting $x=\\simplify{-{d}}$ into $g(x)$ gives</p>\n<p>\\[<br/>\\begin{align}<br/>g(\\simplify{-{d}}) &amp;= \\simplify[all,!collectNumbers]{{coef_x3}+{coef_x2}+{coef_x}+m}\\\\<br/>&amp;=\\simplify{{coef_x3}+{coef_x2}+{coef_x}+m}.<br/>\\end{align}<br/>\\]</p>\n<p>Therefore, as&nbsp;$g(\\simplify{-{d}}) = 0$, we have</p>\n<p>\\[<br/>\\begin{align}<br/>\\simplify{{coef_x3}+{coef_x2}+{coef_x}+m}&amp;=0\\\\<br/>m&amp;=\\simplify{-({coef_x3}+{coef_x2}+{coef_x})}.<br/>\\end{align}<br/>\\]</p>", "variables": {"b": {"name": "b", "definition": "random(-2..3 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": "<p>Random number between -2 and 3 except 0 for creating polynomial.</p>"}, "a": {"name": "a", "definition": "random(-2..3 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": "<p>Random number between -2 and 3, not including 0 for creating polynomial.</p>"}, "coef_x": {"name": "coef_x", "definition": "(a*d+w*b*d+a*b)*(-d)", "group": "Ungrouped variables", "templateType": "anything", "description": "<p>Number obtained by putting x=-d into the third term of the equation.</p>"}, "w": {"name": "w", "definition": "random(2,3,4)", "group": "Ungrouped variables", "templateType": "anything", "description": "<p>Random number between 2,3,4.</p>"}, "coef_x2": {"name": "coef_x2", "definition": "(w*d+a+w*b)*(-d)^2", "group": "Ungrouped variables", "templateType": "anything", "description": "<p>Number obtained by putting x=-d into the second term of the equation.</p>"}, "d": {"name": "d", "definition": "random(-2..2 except 0 except a except b)", "group": "Ungrouped variables", "templateType": "anything", "description": "<p>Used in creation of the polynomial.</p>"}, "coef_x3": {"name": "coef_x3", "definition": "(w)*(-d)^3", "group": "Ungrouped variables", "templateType": "anything", "description": "<p>Number obtained by putting x=-d into the first term of the equation.</p>"}}, "rulesets": {}, "tags": [], "preamble": {"js": "", "css": ""}, "extensions": [], "name": "Factor Theorem and polynomials", "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": ["w", "a", "b", "d", "coef_x3", "coef_x2", "coef_x"], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "showCorrectAnswer": true, "gaps": [{"checkingaccuracy": 0.001, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "checkingtype": "absdiff", "answer": "{-({w}*({-d})^3+({w}*{d}+{a}+{w}*{b})*({-d})^2+({a}*{d}+{w}*{b}*{d}+{a}*{b})*{-d})}", "variableReplacements": [], "expectedvariablenames": [], "type": "jme", "showFeedbackIcon": true, "marks": "2", "vsetrangepoints": 5, "scripts": {}, "showpreview": true, "vsetrange": [0, 1]}], "variableReplacements": [], "scripts": {}, "prompt": "<p>Given that $(\\simplify{x+{d}})$ is a factor of $g(x) =&nbsp;\\simplify{{w}*x^3+({w}{d}+{a}+{w}{b})*x^2+({a}{d}+{w}{b}{d}+{a}{b})*x}+m$, find the value of $m$.</p>\n<p>$m =$ [[0]].</p>\n<p></p>"}], "type": "question", "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}]}]}], "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}]}