// Numbas version: exam_results_page_options
{"name": "Finding the missing value of a constant in a polynomial, using the Factor Theorem ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Finding the missing value of a constant in a polynomial, using the Factor Theorem ", "type": "question", "tags": ["Factor Theorem", "factor theorem", "polynomials", "Polynomials", "taxonomy"], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"coef_x3": {"description": "<p>Number obtained by putting x=-d into the first term of the equation.</p>", "group": "Ungrouped variables", "templateType": "anything", "name": "coef_x3", "definition": "(w)*(-d)^3"}, "b": {"description": "<p>Random number between -2 and 3 except 0 for creating polynomial.</p>", "group": "Ungrouped variables", "templateType": "anything", "name": "b", "definition": "random(-2..3 except 0)"}, "coef_x": {"description": "<p>Number obtained by putting x=-d into the third term of the equation.</p>", "group": "Ungrouped variables", "templateType": "anything", "name": "coef_x", "definition": "(a*d+w*b*d+a*b)*(-d)"}, "d": {"description": "<p>Used in creation of the polynomial.</p>", "group": "Ungrouped variables", "templateType": "anything", "name": "d", "definition": "random(-2..2 except 0 except a except b)"}, "w": {"description": "<p>Random number between 2,3,4.</p>", "group": "Ungrouped variables", "templateType": "anything", "name": "w", "definition": "random(2,3,4)"}, "coef_x2": {"description": "<p>Number obtained by putting x=-d into the second term of the equation.</p>", "group": "Ungrouped variables", "templateType": "anything", "name": "coef_x2", "definition": "(w*d+a+w*b)*(-d)^2"}, "a": {"description": "<p>Random number between -2 and 3, not including 0 for creating polynomial.</p>", "group": "Ungrouped variables", "templateType": "anything", "name": "a", "definition": "random(-2..3 except 0)"}}, "extensions": [], "functions": {}, "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>", "variable_groups": [], "parts": [{"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "type": "jme", "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetrangepoints": 5, "checkingaccuracy": 0.001, "variableReplacements": [], "marks": "2", "expectedvariablenames": [], "answer": "{-({w}*({-d})^3+({w}*{d}+{a}+{w}*{b})*({-d})^2+({a}*{d}+{w}*{b}*{d}+{a}*{b})*{-d})}", "checkvariablenames": false, "showpreview": true}], "showFeedbackIcon": true, "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": "gapfill"}], "ungrouped_variables": ["w", "a", "b", "d", "coef_x3", "coef_x2", "coef_x"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given a factor of a cubic polynomial,&nbsp;factorise it fully by first dividing by the given factor, then factorising the remaining quadratic.</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>", "contributors": [{"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}]}]}], "contributors": [{"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}]}