// Numbas version: exam_results_page_options
{"name": "Encontrar el resto cuando $f(x) = ax^3+bx^2+cx+d$ es dividido por $({{a}x+{k}})$", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Encontrar el resto cuando $f(x) = ax^3+bx^2+cx+d$ es dividido por $({{a}x+{k}})$", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "<p>El teorema del resto establece que si un polinomio $ f(x) $ se divide por $ (\\simplify {a * x-b}) $, entonces el resto es $ f\\left (\\frac {b}{a}\\right) $.</p>\n<p>Esto significa que si sustituimos $ x = \\frac{b}{a}$ en la ecuaci&oacute;n por $ f(x) $, el resultado ser&aacute; igual al resto cuando $ f(x) $ se divide por $ (\\simplify{a * x-b})$.</p>\n<p>Por lo tanto, para calcular el resto cuando $ f(x) = \\simplify {{coef_x3} * x^3 + {coef_x2} * x^2 + {coef_x} * x + {const}} $ se divide por $ (\\simplify { {a}* x+{k}}) $, usamos este mismo principio.</p>\n<p>Como estamos dividiendo $ f(x) $ por $ (\\simplify {{a} * x + {k}}) $, usando el teorema del resto podemos sustituir:</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>en nuestra ecuaci&oacute;n para $ f (x) $ nos dar&aacute; el resto cuando $ f(x) $ se divide por $ (\\simplify {{a}*x+{k}}) $. Sustituir este valor de $x$ en $ f(x)$ nos da</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>Por lo tanto, el resto cuando $f(x) = \\simplify{{coef_x3}*x^3+{coef_x2}*x^2+{coef_x}*x+{const}}$ es dividido por &nbsp;$(\\simplify{{a}*x+{k}})$ es &nbsp;$\\displaystyle\\simplify[all,fractionnumbers]{{coef_x3*(-({k}/{a}))^3}+{coef_x2*(-({k}/{a}))^2}+{coef_x*(-({k}/{a}))}+{const}}$.</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..3 except k except -k)", "description": "<p>Leading coefficient in the dividing equation.</p>", "templateType": "anything", "can_override": false}, "coef_x": {"name": "coef_x", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "<p>Coefficient of x.</p>", "templateType": "anything", "can_override": false}, "remainder": {"name": "remainder", "group": "Ungrouped variables", "definition": "coef_x3*(-k/a)^3+coef_x2*(-k/a)^2+coef_x*(-k/a)+const", "description": "<p>Correct remainder.</p>", "templateType": "anything", "can_override": false}, "coef_x2": {"name": "coef_x2", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "<p>Coefficient of x^2.</p>", "templateType": "anything", "can_override": false}, "coef_x3": {"name": "coef_x3", "group": "Ungrouped variables", "definition": "random(2..4)  ", "description": "<p>Coefficient of x^3</p>", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(-3..3 except 0 except 1 except -1)", "description": "<p>Free&nbsp;coefficient in the dividing equation.</p>", "templateType": "anything", "can_override": false}, "const": {"name": "const", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "<p>Constant term</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["coef_x3", "coef_x2", "coef_x", "const", "k", "remainder", "a"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Encontrar el resto cuando $f(x) = \\simplify{{coef_x3}x^3+{coef_x2}x^2+{coef_x}x+{const}}$ es dividido por $(\\simplify{{a}x+{k}})$,&nbsp;</p>", "minValue": "remainder", "maxValue": "remainder", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "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": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "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": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}