// Numbas version: exam_results_page_options
{"name": "Remainder Theorem", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"js": "", "css": ""}, "ungrouped_variables": ["count", "a_root1", "a_root2", "a_root3", "a_rmndr", "a_coeff2", "a_coeff1", "a_coeff0", "a_q1", "a_q0", "b_root1", "b_root2", "b_root3", "b_coeff2", "b_coeff1", "b_coeff0", "c_root1", "c_root2", "c_root3", "c_r", "c_q1", "c_q0", "c_coeff2", "c_coeff1", "c_coeff0", "d_root1", "d_root2", "d_root3", "d_r", "d_coeff3", "d_coeff2", "d_coeff1", "d_coeff0", "e_root1", "e_root2", "e_root3", "e_r", "e_coeff3", "e_coeff2", "e_coeff1", "e_coeff0", "q_e0", "q_e1", "f0", "f_r", "f_div", "f_p", "g1", "n_root1", "n_root2", "n_root3", "n_r", "n_coeff3", "n_coeff2", "n_coeff1", "n_coeff0"], "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>", "metadata": {"description": "<p>Quotient and remainder, polynomial division.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "prompt": "<p>By using the Polynomial Remainder Theorem, find the remainder $R$ when $\\simplify{f(x)=x^3+{c_coeff2}x^2+{c_coeff1}x+({c_coeff0}+ {c_r})}$ is divided by $\\simplify{(x-{c_root1})}$</p>\n<p>$R = f($[[0]]$)=$[[1]]</p>", "unitTests": [], "gaps": [{"customName": "", "answer": "{c_root1}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": []}, {"customName": "", "answer": "{c_r}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": []}], "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true}, {"customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "prompt": "<p>Find the remainder $R$ when&nbsp;$\\simplify{f(x)={n_coeff3}x^3+{n_coeff2}x^2+{n_coeff1}x+({n_coeff0}+ {n_r})}$ is divided by $\\simplify{({n_coeff3}x-{n_root1})}$</p>\n<p>$R =$ [[0]]&nbsp;</p>\n<p></p>", "unitTests": [], "gaps": [{"customName": "", "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "type": "numberentry", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "useCustomName": false, "allowFractions": true, "variableReplacements": [], "unitTests": [], "minValue": "{n_r}", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{n_r}"}], "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true}, {"customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "prompt": "<p>Find 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>", "unitTests": [], "gaps": [{"customName": "", "answer": "x^2 + {q_e1}x + {q_e0}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": [{"value": "", "name": "x"}]}, {"customName": "", "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "type": "numberentry", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "useCustomName": false, "allowFractions": true, "variableReplacements": [], "unitTests": [], "minValue": "{e_r}", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{e_r}"}], "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true}, {"customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "prompt": "<p>When $\\simplify{f(x)=x^3+px^2+px+{f0}}$ is divided by $\\simplify{x-{f_div}}$ the remainder is $\\var{f_r}$. Find the&nbsp;value of p by setting up an appropriate equation</p>\n<p>$f($[[0]]$)=$[[1]]$^3+$[[2]]$^2p+$[[3]]$p+$[[4]]$=$[[5]]</p>\n<p>Hence $p =$[[6]]</p>", "unitTests": [], "gaps": [{"customName": "", "answer": "{f_div}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": []}, {"customName": "", "answer": "{f_div}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": []}, {"customName": "", "answer": "{f_div}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": "0.5", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": []}, {"customName": "", "answer": "{f_div}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": "0.5", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": []}, {"customName": "", "answer": "{f0}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": "0.5", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": []}, {"customName": "", "answer": "{f_r}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": "0.5", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": []}, {"customName": "", "answer": "{f_p}", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "checkingAccuracy": 0.001, "type": "jme", "showPreview": true, "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "unitTests": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "failureRate": 1, "valuegenerators": []}], "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true}], "tags": [], "statement": "", "variables": {"a_root2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5)", "name": "a_root2"}, "g1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-19..19 except 0)", "name": "g1"}, "b_root2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "b_root2"}, "e_root1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except e_coeff3)", "name": "e_root1"}, "c_root1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "c_root1"}, "a_coeff1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "a_root1*a_root2+a_root2*a_root3+a_root1*a_root3", "name": "a_coeff1"}, "q_e0": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "e_root2*e_root3 ", "name": "q_e0"}, "n_root1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except n_coeff3)", "name": "n_root1"}, "c_coeff1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "c_root1*c_root2 + c_root2*c_root3 + c_root1*c_root3", "name": "c_coeff1"}, "a_q0": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "a_root2*a_root3", "name": "a_q0"}, "n_r": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "name": "n_r"}, "a_coeff0": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "a_root1*a_root2*a_root3", "name": "a_coeff0"}, "c_coeff0": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "(-1)*c_root1*c_root2*c_root3", "name": "c_coeff0"}, "c_q1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "-c_root2-c_root3", "name": "c_q1"}, "n_coeff2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "-(n_coeff3*n_root2 + n_coeff3*n_root3 + n_root1)", "name": "n_coeff2"}, "d_root2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5)", "name": "d_root2"}, "n_root2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5)", "name": "n_root2"}, "e_coeff0": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "e_root1*e_root2*e_root3*(-1)", "name": "e_coeff0"}, "b_root3": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5 except b_root1 except 0)", "name": "b_root3"}, "e_coeff2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "-(e_coeff3*e_root2 + e_coeff3*e_root3 + e_root1)", "name": "e_coeff2"}, "n_coeff1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "n_root1*n_root2 + n_root1*n_root3 + n_coeff3*n_root2*n_root3", "name": "n_coeff1"}, "c_q0": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "c_root2*c_root3", "name": "c_q0"}, "a_root1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5)", "name": "a_root1"}, "a_q1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "-a_root2-a_root3", "name": "a_q1"}, "e_coeff3": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(2..4)", "name": "e_coeff3"}, "a_root3": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5)", "name": "a_root3"}, "e_root3": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5)", "name": "e_root3"}, "d_root1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "d_root1"}, "q_e1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "-e_root2-e_root3", "name": "q_e1"}, "f_r": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-9..9)", "name": "f_r"}, "n_coeff3": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(2..4)", "name": "n_coeff3"}, "c_root2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5 except c_root1)", "name": "c_root2"}, "b_coeff0": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "b_root1*b_root2*b_root3", "name": "b_coeff0"}, "b_coeff2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "b_root1+b_root2+b_root3", "name": "b_coeff2"}, "count": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "0", "name": "count"}, "n_root3": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5)", "name": "n_root3"}, "e_coeff1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "e_root1*e_root2 + e_root1*e_root3 + e_coeff3*e_root2*e_root3", "name": "e_coeff1"}, "n_coeff0": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "n_root1*n_root2*n_root3*(-1)", "name": "n_coeff0"}, "d_coeff1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "d_root1*d_root2/2+ d_root1*d_root3 + d_coeff3*d_root2*d_root3/2", "name": "d_coeff1"}, "f_div": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except -1)", "name": "f_div"}, "f0": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-9..9)", "name": "f0"}, "d_coeff2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "-(d_coeff3*d_root2/2 + d_coeff3*d_root3 + d_root1)", "name": "d_coeff2"}, "c_r": {"templateType": "anything", "description": "<p>Remainder</p>", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "name": "c_r"}, "c_root3": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except c_root1 except c_root2)", "name": "c_root3"}, "d_root3": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5)", "name": "d_root3"}, "e_r": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "name": "e_r"}, "e_root2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5)", "name": "e_root2"}, "a_coeff2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "a_root1+a_root2+a_root3", "name": "a_coeff2"}, "c_coeff2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "-c_root1-c_root2-c_root3", "name": "c_coeff2"}, "a_rmndr": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5)", "name": "a_rmndr"}, "d_coeff0": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "d_root1*d_root2/2*d_root3*(-1)", "name": "d_coeff0"}, "b_root1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "b_root1"}, "f_p": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "(f_r-f0-(f_div)^3)/((f_div)+(f_div)^2)", "name": "f_p"}, "d_coeff3": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "name": "d_coeff3"}, "d_r": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-9..9)", "name": "d_r"}, "b_coeff1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "b_root1*b_root2+b_root2*b_root3+b_root1*b_root3", "name": "b_coeff1"}}, "rulesets": {}, "functions": {}, "extensions": [], "name": "Remainder Theorem", "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/"}]}]}], "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/"}]}