// Numbas version: exam_results_page_options
{"name": "Apply the factor and remainder theorems", "metadata": {"description": "<p>Apply the factor and remainder theorems to manipulate polynomial expressions</p>", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], []], "questions": [{"name": "Use the factor theorem to identify factors of a polynomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "rulesets": {}, "functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "coef1_x3", "coef1_x2", "coef1_x", "const", "coef2_x3", "coef2_x2", "coef2_x", "coef3_x3", "coef3_x2", "coef3_x"], "metadata": {"description": "<p>Apply the&nbsp;factor theorem to check which of a list of linear polynomials are factors of another polynomial.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [], "advice": "<p>To find the factors of the polynomial $f(x) = \\simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}$, we use the factor theorem.&nbsp;</p>\n<p>If $f(x)$ is a polynomial and $f(p) = 0$, then $(x-p)$ is a factor of $f(x)$.</p>\n<p>If $(\\simplify{(x+{a})})$ is a factor of $f(x)$ then by&nbsp;the factor theorem,&nbsp;$f(\\simplify{-{a}}) = 0$.</p>\n<p>We see that</p>\n<p>\\[<br/>\\begin{align}<br/>f(\\simplify{-{a}})&nbsp;&amp;= \\simplify[all,!collectNumbers]{{coef1_x3}+{coef1_x2}+{coef1_x}+{const}}\\\\<br/>&amp;= \\simplify{{coef1_x3}+{coef1_x2}+{coef1_x}+{const}}.<br/>\\end{align}<br/>\\]</p>\n<p>Therefore, $(\\simplify{(x+{a})})$ is a factor of $f(x)$.</p>\n<p>Similarly for $(\\simplify{(x+{d})})$,</p>\n<p>\\[<br/>\\begin{align}<br/>f(\\simplify{-{d}})&nbsp;&amp;= \\simplify[all,!collectNumbers]{{coef2_x3}+{coef2_x2}+{coef2_x}+{const}}\\\\<br/>&amp;= \\simplify{{coef2_x3}+{coef2_x2}+{coef2_x}+{const}}\\\\<br/>&amp;\\neq 0.<br/>\\end{align}<br/>\\]</p>\n<p>Therefore, $(\\simplify{(x+{d})})$ is not a factor of $f(x)$.</p>\n<p>Finally, for $(\\simplify{(x+{c})})$,</p>\n<p>\\[<br/>\\begin{align}<br/>f(\\simplify{-{c}}) &amp;=&nbsp;\\simplify[all,!collectNumbers]{{coef3_x3}+{coef3_x2}+{coef3_x}+{const}}\\\\<br/>&amp;= \\simplify{{coef3_x3}+{coef3_x2}+{coef3_x}+{const}}.<br/>\\end{align}<br/>\\]</p>\n<p>Therefore, $(\\simplify{(x+{c})})$ is also a factor of $f(x)$.</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>", "preamble": {"js": "", "css": ""}, "variables": {"coef1_x3": {"templateType": "anything", "description": "<p>Number obtained from putting x=-a into the first term of the equation.</p>", "name": "coef1_x3", "group": "Ungrouped variables", "definition": "(-a)^3"}, "coef1_x2": {"templateType": "anything", "description": "<p>Number obtained from putting x=-a into the second term of the equation.</p>", "name": "coef1_x2", "group": "Ungrouped variables", "definition": "(a+b+c)*(-a)^2"}, "coef2_x3": {"templateType": "anything", "description": "<p>Number obtained from putting x=-d into the&nbsp;first term&nbsp;in the&nbsp;equation.</p>", "name": "coef2_x3", "group": "Ungrouped variables", "definition": "(-d)^3"}, "coef1_x": {"templateType": "anything", "description": "<p>Number obtained from putting x=-a into the first term of the equation.</p>", "name": "coef1_x", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-a)"}, "coef3_x3": {"templateType": "anything", "description": "<p>Number obtained for putting x=-c into the first term of the equation.</p>", "name": "coef3_x3", "group": "Ungrouped variables", "definition": "(-c)^3"}, "const": {"templateType": "anything", "description": "<p>Constant term in the equation.</p>", "name": "const", "group": "Ungrouped variables", "definition": "a*b*c"}, "c": {"templateType": "anything", "description": "<p>Random number between -2 and 3 except 0 for creating polynomial.</p>", "name": "c", "group": "Ungrouped variables", "definition": "random(-2..3 except 0)"}, "d": {"templateType": "anything", "description": "<p>Incorrect answer for part a.</p>", "name": "d", "group": "Ungrouped variables", "definition": "random(-2..2 except 0 except a except c except b)"}, "coef2_x": {"templateType": "anything", "description": "<p>Number obtained from putting x=-d into the 3rd term for the equation.</p>", "name": "coef2_x", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-d)"}, "a": {"templateType": "anything", "description": "<p>Random number between -2 and 3, not including 0 for creating polynomial.</p>", "name": "a", "group": "Ungrouped variables", "definition": "random(-2..3 except 0 except c)"}, "coef2_x2": {"templateType": "anything", "description": "<p>Number obtained from putting x=-d into the second term of the equation.</p>", "name": "coef2_x2", "group": "Ungrouped variables", "definition": "(a+b+c)*(-d)^2"}, "b": {"templateType": "anything", "description": "<p>Random number between -2 and 3 except 0 for creating polynomial.</p>", "name": "b", "group": "Ungrouped variables", "definition": "random(-2..3 except 0 except c)"}, "coef3_x": {"templateType": "anything", "description": "<p>Number obtained by putting x=-c into the third term of the equation.</p>", "name": "coef3_x", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-c)"}, "coef3_x2": {"templateType": "anything", "description": "", "name": "coef3_x2", "group": "Ungrouped variables", "definition": "(a+b+c)*(-c)^2"}}, "parts": [{"variableReplacementStrategy": "originalfirst", "choices": ["<p>$(\\simplify{x+{a}})$</p>", "<p>$(\\simplify{x+{d}})$</p>", "<p>$(\\simplify{x+{c}})$</p>"], "matrix": ["1", 0, "1"], "displayColumns": 0, "distractors": ["", "", ""], "maxMarks": 0, "type": "m_n_2", "minMarks": 0, "marks": 0, "maxAnswers": 0, "prompt": "<p>Use the factor theorem to find&nbsp;which <strong>two</strong> of the following are&nbsp;factors of the&nbsp;polynomial</p>\n<p>\\[f(x) = \\simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}.\\]</p>", "variableReplacements": [], "warningType": "none", "showFeedbackIcon": true, "displayType": "checkbox", "shuffleChoices": true, "showCorrectAnswer": true, "minAnswers": 0, "scripts": {}}], "tags": ["factor theorem", "Factor Theorem", "factors", "Factors", "Multiple choice", "Multiple Choice", "multiple choice", "polynomial", "Polynomial", "taxonomy"], "variablesTest": {"maxRuns": 100, "condition": ""}}, {"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}, "contributors": [{"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "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)"}}, "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>"}, {"name": "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}, "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/"}], "type": "question", "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>", "variablesTest": {"condition": "", "maxRuns": "192"}, "variables": {"y": {"description": "<p>Factor 3.</p>", "definition": "random(-2..3 except 0)", "templateType": "anything", "name": "y", "group": "Ungrouped variables"}, "u": {"description": "<p>Factor 2.</p>", "definition": "random(-2..3 except 0 except -y)", "templateType": "anything", "name": "u", "group": "Ungrouped variables"}, "z": {"description": "<p>Factor 1.</p>", "definition": "random(-3..3 except 0)", "templateType": "anything", "name": "z", "group": "Ungrouped variables"}}, "functions": {}, "tags": ["factor theorem", "Factor Theorem", "factorisation", "Factorisation", "factorise", "Factorise", "Long division", "long division", "polynomials", "Polynomials", "taxonomy"], "variable_groups": [], "parts": [{"extendBaseMarkingAlgorithm": true, "vsetRange": [0, 1], "scripts": {}, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "showFeedbackIcon": true, "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>", "failureRate": 1, "answer": "(x+{y})(x+{u})(x+{z})", "marks": "2", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "vsetRangePoints": 5, "unitTests": [], "valuegenerators": [{"value": "", "name": "x"}], "showPreview": true, "checkVariableNames": false, "mustmatchpattern": {"message": "Your answer is not fully factorised.", "pattern": "((`+-x^$n`? + `+- $n)^$n`?)`* * $z", "partialCredit": "50", "nameToCompare": ""}, "customMarkingAlgorithm": "", "useCustomName": false, "customName": "", "variableReplacements": []}], "ungrouped_variables": ["y", "u", "z"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Use a given factor of a polynomial to find the full factorisation of the polynomial through long division.</p>"}, "preamble": {"css": "", "js": ""}, "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>"}, {"name": "Finding unknown coefficients of a polynomial by  the remainder theorem", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "tags": [], "metadata": {"description": "<p>This question tests the student's knowledge of the remainder theorem and the ways in which it can be applied.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Consider the polynomial</p>\n<p>\\[ p(x) = \\simplify{{coef2_x3}x^3+s*x^2+{coef2_x}x+t}\\text{.}\\]</p>\n<p>The polynomial:</p>\n<ul>\n<li>has remainder $\\var{rem1}$ when divided by $(\\simplify{x+{c}})$</li>\n<li>has remainder $\\var{rem2}$ when divided by $(\\simplify{x+{d}})$</li>\n</ul>", "advice": "<p>We are told that the polynomial:</p>\n<ul>\n<li>has a remainder $\\var{rem1}$ when divided by $(\\simplify{x+{c}})$</li>\n<li>has&nbsp;a remainder $\\var{rem2}$ when divided by $(\\simplify{x+{d}})$</li>\n</ul>\n<h4>a)</h4>\n<p>Firstly, substituting $x = \\simplify{-{c}}$ into $p(x)$ gives us&nbsp;</p>\n<p>\\begin{align}<br/>p(\\simplify{-{c}}) &amp;= \\simplify[all,!collectNumbers, fractionnumbers]{{coef2_x3*(-c)^3}+{(-c)^2}s+{coef2_x*(-c)}+t},\\\\<br/>&amp;= \\simplify[all,fractionnumbers]{{coef2_x3*(-c)^3}+{(-c)^2}s+{coef2_x*(-c)}+t}.<br/>\\end{align}</p>\n<p>But, by the remainder theorem $p(\\simplify{-{c}}) = \\var{rem1}$ (using the first bullet point), so&nbsp;this becomes</p>\n<p>\\begin{align}<br/>\\simplify[all,fractionnumbers]{{coef2_x3*(-{c})^3}+s*{(-{c})^2}+{coef2_x*(-{c})}+t} &amp;= \\var{rem1},\\\\<br/>\\simplify[all,fractionnumbers]{s*{x}+t} &amp;= \\simplify[all,fractionnumbers]{{rem1}-{coef2_x3*(-{c})^3}-{coef2_x*(-{c})}}.<br/>\\end{align}</p>\n<h4>b)</h4>\n<p>Similarly, substituting $x = \\simplify{-{d}}$ into $p(x)$, gives us</p>\n<p>\\begin{align}<br/>p(\\simplify{-{d}}) &amp;= \\simplify[all,!collectNumbers, fractionnumbers]{{coef2_x3*(-{d})^3}+{(-{d})^2}s+{coef2_x*(-{d})}+t},\\\\<br/>&amp;= \\simplify[all,fractionnumbers]{{coef2_x3*(-{d})^3}+{(-{d})^2}s+{coef2_x*(-{d})}+t}.<br/>\\end{align}</p>\n<p>But, by the remainder theorem $p(\\simplify{-{d}}) = \\var{rem2}$ (using the second bullet point), so this becomes</p>\n<p>\\begin{align}<br/>\\simplify[all,fractionnumbers]{{coef2_x3*(-{d})^3}+s*{(-{d})^2}+{coef2_x*(-{d})}+t} &amp;= \\var{rem2},\\\\<br/>\\simplify[all,fractionnumbers]{s*{y}+t} &amp;= \\simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}.<br/>\\end{align}</p>\n<h4>c)</h4>\n<p>We now have two simultaneous equations for $s$ and $t$:</p>\n<p>\\begin{align}<br/>\\simplify[all,fractionnumbers]{s*{x}+t} = \\simplify[all,fractionnumbers]{{rem1}-{coef2_x3*(-{c})^3}-{coef2_x*(-{c})}} \\\\<br/>\\simplify[all,fractionnumbers]{s*{y}+t} = \\simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}<br/>\\end{align}</p>\n<p>Next, we subtract the&nbsp;second equation from the&nbsp;first equation.</p>\n<p>This allows us to cancel out the terms involving $t$ and gives us an equation only in terms of $s$, which we can then rearrange to find the value of $s$.</p>\n<p>Subtracting the two&nbsp;equations gives</p>\n<p>\\[\\simplify{s*{(-{c})^2-(-{d})^2}} = \\simplify[all,fractionnumbers]{{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}}.\\]</p>\n<p>Then, we can rearrange this equation so that</p>\n<p>\\[s = \\simplify[all,fractionnumbers]{{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}/{{(-c)^2-(-d)^2}}}.\\]</p>\n<h4>d)</h4>\n<p>We can calculate $t$ by substituting our value of $s$ into one of our original simultaneous equations. For example,&nbsp;let's use the&nbsp;equation</p>\n<p>\\[\\simplify[all,fractionnumbers]{s*{(-{d})^2}+t} = \\simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}.\\]</p>\n<p>Substituting our value of $s$ into this equation gives us</p>\n<p>\\[<br/>\\begin{align}<br/>\\simplify[all,fractionnumbers,!noleadingMinus]{{numerator/denominator}+t} &amp;= \\simplify[all,fractionnumbers]{{rem2-coef2_x3*(-d)^3-coef2_x*(-d)}},\\\\<br/>t &amp;= \\simplify[all,fractionnumbers]{{rem2-coef2_x3*(-d)^3-coef2_x*(-d)&nbsp;- numerator/denominator}}.<br/>\\end{align}<br/>\\]</p>\n<p>This same answer would've also been obtained if we had substituted our value of $s$ into the other equation instead.</p>", "rulesets": {}, "variables": {"numerator": {"name": "numerator", "group": "Ungrouped variables", "definition": "rem1 - rem2+coef2_x3*(c^3-d^3)+coef2_x*(c-d)", "description": "<p>Numerator of s</p>", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-3..3 except [0,1])", "description": "<p>Dividing term 1.</p>", "templateType": "anything"}, "rem1": {"name": "rem1", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "<p>First remainder.</p>", "templateType": "anything"}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "numerator/denominator", "description": "", "templateType": "anything"}, "coef2_x": {"name": "coef2_x", "group": "Ungrouped variables", "definition": "random(-3..3 except [0,c,d])", "description": "<p>Coefficient of x.</p>", "templateType": "anything"}, "denominator": {"name": "denominator", "group": "Ungrouped variables", "definition": "c^2-d^2", "description": "<p>Denominator of s.</p>", "templateType": "anything"}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "(rem2-coef2_x3*(-d)^3-coef2_x*(-d) - (rem1 - rem2-coef2_x3*(-c)^3-coef2_x*(-c)+coef2_x3*(-d)^3+coef2_x*(-d))/((-c)^2-(-d)^2)*(-d)^2)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-3..3 except c except [0,1,-c])", "description": "<p>Dividing term 2.</p>", "templateType": "anything"}, "coef2_x3": {"name": "coef2_x3", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "<p>Coefficient of x^3.</p>", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(-(c))^2", "description": "<p>Simplifies first coefficient of s.</p>", "templateType": "anything"}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "(-(d))^2", "description": "<p>Simplifies second coefficient of s.</p>", "templateType": "anything"}, "rem2": {"name": "rem2", "group": "Ungrouped variables", "definition": "random(-3..3 except 0) ", "description": "<p>Second remainder.</p>", "templateType": "anything"}}, "variablesTest": {"condition": "x > y", "maxRuns": 100}, "ungrouped_variables": ["c", "rem1", "d", "rem2", "coef2_x3", "coef2_x", "numerator", "denominator", "s", "x", "y", "t"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Using the remainder theorem for the remainder when $p(x)$ is divided by $(\\simplify{x+{c}})$, create an equation involving&nbsp;$s$ and $t$.</p>\n<p>[[0]]$s + t$ = [[1]].</p>\n<p></p>", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The remainder theorem states that if a polynomial $f(x)$ is divided by $(\\simplify{a*x-b})$ then the remainder is $f(\\frac{b}{a})$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Coefficient of s", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{c}^2", "maxValue": "{c}^2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Right-hand side", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{rem1}+{coef2_x3}*(c^3)+{coef2_x}*(c)", "maxValue": "{rem1}+{coef2_x3}*(c^3)+{coef2_x}*(c)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Using the remainder theorem for the remainder when $p(x)$ is divided by $(\\simplify{x+{d}})$, create another equation involving $s$ and $t$.</p>\n<p>[[0]]$s+t$ = [[1]].</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{d}^2", "maxValue": "{d}^2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{rem2}+{coef2_x3}*(d^3)+{coef2_x}*d", "maxValue": "{rem2}+{coef2_x3}*(d^3)+{coef2_x}*d", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the value of $s$. Reduce your answer to its simplest fractional form.</p>\n<p>$s =$ [[0]]</p>", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Subtract the two simultaneous equations for $s$ and $t$, obtained&nbsp;in parts a) and b), from each other.</p>\n<p>Then rearrange this new equation to find the value of $s$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "s", "maxValue": "s", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the value of $t$. Reduce your answer to its simplest fractional form.</p>\n<p>$t =$ [[0]]</p>", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Substitute the value of $s$ from part c) into one of the simultaneous equations for $s$ and $t$.</p>\n<p>Then, rearrange this equation to find the value of $t$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{(rem2-coef2_x3*(-d)^3-coef2_x*(-d) - (rem1 - rem2-coef2_x3*(-c)^3-coef2_x*(-c)+coef2_x3*(-d)^3+coef2_x*(-d))/((-c)^2-(-d)^2)*(-d)^2)}", "maxValue": "{(rem2-coef2_x3*(-d)^3-coef2_x*(-d) - (rem1 - rem2-coef2_x3*(-c)^3-coef2_x*(-c)+coef2_x3*(-d)^3+coef2_x*(-d))/((-c)^2-(-d)^2)*(-d)^2)}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Dividing a polynomial with remainders, using the remainder theorem", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "type": "question", "statement": "", "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"const": {"group": "Ungrouped variables", "name": "const", "description": "<p>Constant term</p>", "templateType": "anything", "definition": "random(-5..5 except 0)"}, "coef_x2": {"group": "Ungrouped variables", "name": "coef_x2", "description": "<p>Coefficient of x^2.</p>", "templateType": "anything", "definition": "random(-3..3 except 0)"}, "coef_x": {"group": "Ungrouped variables", "name": "coef_x", "description": "<p>Coefficient of x.</p>", "templateType": "anything", "definition": "random(-3..3 except 0)"}, "coef_x3": {"group": "Ungrouped variables", "name": "coef_x3", "description": "<p>Coefficient of x^3</p>", "templateType": "anything", "definition": "random(2..4)  "}, "remainder": {"group": "Ungrouped variables", "name": "remainder", "description": "<p>Correct remainder.</p>", "templateType": "anything", "definition": "coef_x3*(-k/a)^3+coef_x2*(-k/a)^2+coef_x*(-k/a)+const"}, "k": {"group": "Ungrouped variables", "name": "k", "description": "<p>Free&nbsp;coefficient in the dividing equation.</p>", "templateType": "anything", "definition": "random(-3..3 except 0 except 1 except -1)"}, "a": {"group": "Ungrouped variables", "name": "a", "description": "<p>Leading coefficient in the dividing equation.</p>", "templateType": "anything", "definition": "random(1..3 except k except -k)"}}, "tags": ["taxonomy"], "variable_groups": [], "parts": [{"correctAnswerFraction": true, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "remainder", "showFeedbackIcon": true, "prompt": "<p>Find the remainder when $f(x) = \\simplify{{coef_x3}x^3+{coef_x2}x^2+{coef_x}x+{const}}$ is divided by $(\\simplify{{a}x+{k}})$, using the remainder theorem.</p>", "correctAnswerStyle": "plain", "allowFractions": true, "mustBeReduced": true, "minValue": "remainder", "variableReplacements": [], "marks": "2", "scripts": {}, "showCorrectAnswer": true}], "ungrouped_variables": ["coef_x3", "coef_x2", "coef_x", "const", "k", "remainder", "a"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>This question tests the student's ability to find remainders using the remainder theorem.&nbsp;</p>"}, "preamble": {"css": "", "js": ""}, "functions": {}, "advice": "<p>The remainder theorem states that if a polynomial $f(x)$ is divided by $(\\simplify{a*x-b})$ then the remainder is $f \\left( \\frac{b}{a} \\right)$.</p>\n<p>This means that if we substitute $x = \\frac{b}{a}$ into the equation for $f(x)$, the result will be equal to the remainder when $f(x)$ is divided by $(\\simplify{a*x-b})$.</p>\n<p>Therefore, to calculate the remainder when $f(x) = \\simplify{{coef_x3}*x^3+{coef_x2}*x^2+{coef_x}*x+{const}}$ is divided by $(\\simplify{{a}*x+{k}})$, we use this same principle.</p>\n<p>As we are dividing $f(x)$ by $(\\simplify{{a}*x+{k}})$, using the remainder theorem tells us that substituting</p>\n<p>\\[<br/>\\begin{align}<br/>x &amp;= \\frac{b}{a}\\\\<br/>&amp;= \\simplify{-({k}/{a})}&nbsp;<br/>\\end{align}<br/>\\]</p>\n<p>into our equation for $f(x)$ will give us the remainder when $f(x)$ is divided by $(\\simplify{{a}*x+{k}})$. Substituting this value of $x$ into $f(x)$ gives us</p>\n<p>\\[<br/>\\begin{align}<br/>f(\\simplify{-({k}/{a})}) &amp;= \\simplify[all,!collectNumbers, fractionnumbers]{{coef_x3*(-({k}/{a}))^3}+{coef_x2*(-({k}/{a}))^2}+{coef_x*(-({k}/{a}))}+{const}}\\\\<br/>&amp;= \\simplify[all,fractionnumbers]{{coef_x3*(-({k}/{a}))^3}+{coef_x2*(-({k}/{a}))^2}+{coef_x*(-({k}/{a}))}+{const}}.<br/>\\end{align}<br/>\\]&nbsp;</p>\n<p>Therefore, the remainder when $f(x) = \\simplify{{coef_x3}*x^3+{coef_x2}*x^2+{coef_x}*x+{const}}$ is divided by $(\\simplify{{a}*x+{k}})$ is &nbsp;$\\displaystyle\\simplify[all,fractionnumbers]{{coef_x3*(-({k}/{a}))^3}+{coef_x2*(-({k}/{a}))^2}+{coef_x*(-({k}/{a}))}+{const}}$.</p>"}, {"name": "Remainder Theorem", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}, {"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "tags": [], "metadata": {"description": "<p>Quotient and remainder, polynomial division.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Apply the remainder and factor theorems.</p>", "advice": "<p>The polynomial remainder theorem is useful here, it states that the remainder of the division of a polynomial $f(x)$ by a linear polynomial $x-a$ is equal to $f(a)$.</p>\n<p>To find the quotient, here are two methods:</p>\n<p>1) Perform polynomial long division</p>\n<p>2) First find the remainder by the polynomial remainder theorem. Minus the remainder from the initial polynomial, then factorise.&nbsp;</p>", "rulesets": {}, "variables": {"a_root2": {"name": "a_root2", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "g1": {"name": "g1", "group": "Ungrouped variables", "definition": "random(-19..19 except 0)", "description": "", "templateType": "anything"}, "b_root2": {"name": "b_root2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything"}, "e_root1": {"name": "e_root1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except e_coeff3)", "description": "", "templateType": "anything"}, "c_root1": {"name": "c_root1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything"}, "a_coeff1": {"name": "a_coeff1", "group": "Ungrouped variables", "definition": "a_root1*a_root2+a_root2*a_root3+a_root1*a_root3", "description": "", "templateType": "anything"}, "q_e0": {"name": "q_e0", "group": "Ungrouped variables", "definition": "e_root2*e_root3 ", "description": "", "templateType": "anything"}, "n_root1": {"name": "n_root1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except n_coeff3)", "description": "", "templateType": "anything"}, "c_coeff1": {"name": "c_coeff1", "group": "Ungrouped variables", "definition": "c_root1*c_root2 + c_root2*c_root3 + c_root1*c_root3", "description": "", "templateType": "anything"}, "a_q0": {"name": "a_q0", "group": "Ungrouped variables", "definition": "a_root2*a_root3", "description": "", "templateType": "anything"}, "n_r": {"name": "n_r", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything"}, "a_coeff0": {"name": "a_coeff0", "group": "Ungrouped variables", "definition": "a_root1*a_root2*a_root3", "description": "", "templateType": "anything"}, "c_coeff0": {"name": "c_coeff0", "group": "Ungrouped variables", "definition": "(-1)*c_root1*c_root2*c_root3", "description": "", "templateType": "anything"}, "c_q1": {"name": "c_q1", "group": "Ungrouped variables", "definition": "-c_root2-c_root3", "description": "", "templateType": "anything"}, "n_coeff2": {"name": "n_coeff2", "group": "Ungrouped variables", "definition": "-(n_coeff3*n_root2 + n_coeff3*n_root3 + n_root1)", "description": "", "templateType": "anything"}, "d_root2": {"name": "d_root2", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "n_root2": {"name": "n_root2", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "e_coeff0": {"name": "e_coeff0", "group": "Ungrouped variables", "definition": "e_root1*e_root2*e_root3*(-1)", "description": "", "templateType": "anything"}, "b_root3": {"name": "b_root3", "group": "Ungrouped variables", "definition": "random(-5..5 except b_root1 except 0)", "description": "", "templateType": "anything"}, "e_coeff2": {"name": "e_coeff2", "group": "Ungrouped variables", "definition": "-(e_coeff3*e_root2 + e_coeff3*e_root3 + e_root1)", "description": "", "templateType": "anything"}, "n_coeff1": {"name": "n_coeff1", "group": "Ungrouped variables", "definition": "n_root1*n_root2 + n_root1*n_root3 + n_coeff3*n_root2*n_root3", "description": "", "templateType": "anything"}, "c_q0": {"name": "c_q0", "group": "Ungrouped variables", "definition": "c_root2*c_root3", "description": "", "templateType": "anything"}, "a_root1": {"name": "a_root1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "a_q1": {"name": "a_q1", "group": "Ungrouped variables", "definition": "-a_root2-a_root3", "description": "", "templateType": "anything"}, "e_coeff3": {"name": "e_coeff3", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "a_root3": {"name": "a_root3", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "e_root3": {"name": "e_root3", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "d_root1": {"name": "d_root1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything"}, "q_e1": {"name": "q_e1", "group": "Ungrouped variables", "definition": "-e_root2-e_root3", "description": "", "templateType": "anything"}, "f_r": {"name": "f_r", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything"}, "n_coeff3": {"name": "n_coeff3", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "c_root2": {"name": "c_root2", "group": "Ungrouped variables", "definition": "random(-5..5 except c_root1)", "description": "", "templateType": "anything"}, "b_coeff0": {"name": "b_coeff0", "group": "Ungrouped variables", "definition": "b_root1*b_root2*b_root3", "description": "", "templateType": "anything"}, "b_coeff2": {"name": "b_coeff2", "group": "Ungrouped variables", "definition": "b_root1+b_root2+b_root3", "description": "", "templateType": "anything"}, "count": {"name": "count", "group": "Ungrouped variables", "definition": "0", "description": "", "templateType": "anything"}, "n_root3": {"name": "n_root3", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "e_coeff1": {"name": "e_coeff1", "group": "Ungrouped variables", "definition": "e_root1*e_root2 + e_root1*e_root3 + e_coeff3*e_root2*e_root3", "description": "", "templateType": "anything"}, "n_coeff0": {"name": "n_coeff0", "group": "Ungrouped variables", "definition": "n_root1*n_root2*n_root3*(-1)", "description": "", "templateType": "anything"}, "d_coeff1": {"name": "d_coeff1", "group": "Ungrouped variables", "definition": "d_root1*d_root2/2+ d_root1*d_root3 + d_coeff3*d_root2*d_root3/2", "description": "", "templateType": "anything"}, "f_div": {"name": "f_div", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except -1)", "description": "", "templateType": "anything"}, "f0": {"name": "f0", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything"}, "d_coeff2": {"name": "d_coeff2", "group": "Ungrouped variables", "definition": "-(d_coeff3*d_root2/2 + d_coeff3*d_root3 + d_root1)", "description": "", "templateType": "anything"}, "c_r": {"name": "c_r", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "<p>Remainder</p>", "templateType": "anything"}, "c_root3": {"name": "c_root3", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except c_root1 except c_root2)", "description": "", "templateType": "anything"}, "d_root3": {"name": "d_root3", "group": 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"marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Obtain&nbsp;the quotient $Q(x)$ and remainder $R$ when&nbsp;$\\simplify{f(x)={e_coeff3}x^3+{e_coeff2}x^2+{e_coeff1}x+({e_coeff0}+ {e_r})}$ is divided by $\\simplify{({e_coeff3}x-{e_root1})}$</p>\n<p>$Q(x) =$ [[0]]</p>\n<p>$R =$ [[1]]&nbsp;</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Quotient", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, 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