// Numbas version: exam_results_page_options
{"name": "CF Maths Portfolio - Polynomial Division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "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"], "name": "CF Maths Portfolio - Polynomial Division", "tags": [], "preamble": {"css": "", "js": ""}, "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>\n<p></p>\n<p>If you are unsure of the specific steps involved here, start by studying <em>'polynomial long division'</em>: <a href=\"http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-polydiv-2009-1.pdf \">http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-polydiv-2009-1.pdf </a>and&nbsp;<a href=\"https://www.khanacademy.org/math/algebra2/arithmetic-with-polynomials/long-division-of-polynomials/v/polynomial-division\">https://www.khanacademy.org/math/algebra2/arithmetic-with-polynomials/long-division-of-polynomials/v/polynomial-division</a> are both useful resources.</p>\n<p>Next, study <em>'polynomial remainder theorem'</em>: <a href=\"https://www.khanacademy.org/math/algebra2/arithmetic-with-polynomials/polynomial-remainder-theorem/v/polynomial-remainder-theorem\">https://www.khanacademy.org/math/algebra2/arithmetic-with-polynomials/polynomial-remainder-theorem/v/polynomial-remainder-theorem</a></p>", "rulesets": {}, "parts": [{"prompt": "<p>Find the quotient and remainder when $\\simplify{g(x) = x^3 - {a_coeff2}x^2+({a_coeff1})x -({a_coeff0}+{a_rmndr})}$ is divided by $\\simplify{x-{a_root1}}$</p>\n<p>Quotient $=$&nbsp;[[0]]</p>\n<p>Remainder $=$ [[1]]&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "x^2+{a_q1}x+{a_q0}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{a_rmndr}*(-1)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Hence write $g(x)$ in the form $g(x) = Q(x)(\\simplify{(x-{a_root1})}) + R$</p>\n<p>[[0]]$(\\simplify{x-{a_root1}})+$[[1]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "x^2+{a_q1}x+{a_q0}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{a_rmndr}*(-1)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>By evaluating &nbsp;$\\simplify{f(x)=x^3-{b_coeff2}x^2+{b_coeff1}x-{b_coeff0}}$ at an appropriate value of x, show that $\\simplify{x-{b_root1}}$ is a factor of $f(x)$</p>\n<p style=\"text-align: center;\">$f($[[0]]$)=$[[1]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{b_root1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "0", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"notallowed": {"message": "", "showStrings": false, "strings": ["^"], "partialCredit": 0}, "vsetrangepoints": 5, "prompt": "<p>Hence factorise&nbsp;$\\simplify{f(x)=x^3-{b_coeff2}x^2+{b_coeff1}x-{b_coeff0}}$ completely</p>", "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "musthave": {"message": "", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}, "scripts": {}, "answer": "(x-{b_root1})(x-{b_root2})(x-{b_root3})", "marks": 1, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme"}, {"integerPartialCredit": 0, "prompt": "<p>By using the Polynomial Remainder Theorem, find the remainder 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>", "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{c_r}", "minValue": "{c_r}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "prompt": "<p>By using the Polynomial Remainder Theorem, find the remainder when $\\simplify{f(x)={d_coeff3}x^3+{d_coeff2}x^2+{d_coeff1}x+({d_coeff0}+ {d_r})}$ is divided by $\\simplify{(x-{d_root2})}$</p>", "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{d_r}", "minValue": "{d_r}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>Find the quotient and remainder 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>Quotient $=$ [[0]]</p>\n<p>Remainder $=$ [[1]]&nbsp;</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "x^2 + {q_e1}x + {q_e0}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"allowFractions": true, "variableReplacements": [], "maxValue": "{e_r}", "minValue": "{e_r}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"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.</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "{f_p}", "minValue": "{f_p}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "extensions": [], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a_root3": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a_root3", "description": ""}, "a_root2": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a_root2", "description": ""}, "a_root1": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a_root1", "description": ""}, "b_root1": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b_root1", "description": ""}, "c_q1": {"definition": "-c_root2-c_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "c_q1", "description": ""}, "c_q0": {"definition": "c_root2*c_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "c_q0", "description": ""}, "f0": {"definition": "random(-9..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "f0", "description": ""}, "b_root3": {"definition": "random(-5..5 except b_root1 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b_root3", "description": ""}, "b_coeff0": {"definition": "b_root1*b_root2*b_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "b_coeff0", "description": ""}, "b_coeff1": {"definition": "b_root1*b_root2+b_root2*b_root3+b_root1*b_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "b_coeff1", "description": ""}, "b_coeff2": {"definition": "b_root1+b_root2+b_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "b_coeff2", "description": ""}, "q_e1": {"definition": "-e_root2-e_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "q_e1", "description": ""}, "a_coeff1": {"definition": "a_root1*a_root2+a_root2*a_root3+a_root1*a_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "a_coeff1", "description": ""}, "a_coeff0": {"definition": "a_root1*a_root2*a_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "a_coeff0", "description": ""}, "a_coeff2": {"definition": "a_root1+a_root2+a_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "a_coeff2", "description": ""}, "d_coeff2": {"definition": "-(d_coeff3*d_root2 + d_coeff3*d_root3 + d_root1)", "templateType": "anything", "group": "Ungrouped variables", "name": "d_coeff2", "description": ""}, "d_coeff3": {"definition": "random(-3..3 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "d_coeff3", "description": ""}, "d_coeff0": {"definition": "d_root1*d_root2*d_root3*(-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "d_coeff0", "description": ""}, "d_coeff1": {"definition": "d_root1*d_root2 + d_root1*d_root3 + d_coeff3*d_root2*d_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "d_coeff1", "description": ""}, "f_p": {"definition": "(f_r-f0-(f_div)^3)/((f_div)+(f_div)^2)", "templateType": "anything", "group": "Ungrouped variables", "name": "f_p", "description": ""}, "a_q1": {"definition": "-a_root2-a_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "a_q1", "description": ""}, "a_q0": {"definition": "a_root2*a_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "a_q0", "description": ""}, "c_coeff2": {"definition": "-c_root1-c_root2-c_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "c_coeff2", "description": ""}, "b_root2": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b_root2", "description": ""}, "c_coeff0": {"definition": "(-1)*c_root1*c_root2*c_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "c_coeff0", "description": ""}, "e_coeff1": {"definition": "e_root1*e_root2 + e_root1*e_root3 + e_coeff3*e_root2*e_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "e_coeff1", "description": ""}, "e_coeff0": {"definition": "e_root1*e_root2*e_root3*(-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "e_coeff0", "description": ""}, "e_coeff3": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "e_coeff3", "description": ""}, "e_coeff2": {"definition": "-(e_coeff3*e_root2 + e_coeff3*e_root3 + e_root1)", "templateType": "anything", "group": "Ungrouped variables", "name": "e_coeff2", "description": ""}, "e_root3": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "e_root3", "description": ""}, "e_root2": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "e_root2", "description": ""}, "e_root1": {"definition": "random(-5..5 except 0 except e_coeff3)", "templateType": "anything", "group": "Ungrouped variables", "name": "e_root1", "description": ""}, "c_coeff1": {"definition": "c_root1*c_root2 + c_root2*c_root3 + c_root1*c_root3", "templateType": "anything", "group": "Ungrouped variables", "name": "c_coeff1", "description": ""}, "c_root3": {"definition": "random(-5..5 except 0 except c_root1 except c_root2)", "templateType": "anything", "group": "Ungrouped variables", "name": "c_root3", "description": ""}, "c_r": {"definition": "random(-9..9 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c_r", "description": "<p>Remainder</p>"}, "c_root2": {"definition": "random(-5..5 except c_root1)", "templateType": "anything", "group": "Ungrouped variables", "name": "c_root2", "description": ""}, "count": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "count", "description": ""}, "e_r": {"definition": "random(-9..9 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "e_r", "description": ""}, "a_rmndr": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a_rmndr", "description": ""}, "d_root2": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d_root2", "description": ""}, "d_root3": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d_root3", "description": ""}, "d_root1": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "d_root1", "description": ""}, "d_r": {"definition": "random(-9..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "d_r", "description": ""}, "f_div": {"definition": "random(-5..5 except 0 except -1)", "templateType": "anything", "group": "Ungrouped variables", "name": "f_div", "description": ""}, "c_root1": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c_root1", "description": ""}, "f_r": {"definition": "random(-9..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "f_r", "description": ""}, "q_e0": {"definition": "e_root2*e_root3 ", "templateType": "anything", "group": "Ungrouped variables", "name": "q_e0", "description": ""}, "g1": {"definition": "random(-19..19 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "g1", "description": ""}}, "metadata": {"description": "<p>Quotient and remainder, polynomial division.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}]}]}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}]}