// Numbas version: exam_results_page_options
{"name": "Considere el polinomio $p(x)$ con parametros $t$ y $s$. $ p(x) = ax^3+sx^2+t$, halle los valores de $s$ y $t$.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Considere el polinomio $p(x)$ con parametros $t$ y $s$. $ p(x) = ax^3+sx^2+t$, halle los valores de $s$ y $t$.", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Considere el polinomio $p(x)$ con parametros $t$ y $s$. $ p(x) = \\simplify{{coef2_x3}x^3+s*x^2+{coef2_x}x+t}\\text{.}$</p>\n<p>El polinomio $p(x)$:</p>\n<ul>\n<li>tiene resto $\\var{rem1}$ cuando se divide por $(\\simplify{x+{c}})$</li>\n<li>tiene resto $\\var{rem2}$ cuando se divide por $(\\simplify{x+{d}})$</li>\n</ul>", "advice": "<p>Nos dicen que el polinomio:</p>\n<ul>\n<li>tiene resto $\\var{rem1}$ cuando se divide por $(\\simplify{x+{c}})$</li>\n<li>tiene resto $\\var{rem2}$ cuando se divide por $(\\simplify{x+{d}})$</li>\n</ul>\n<h4>a)</h4>\n<p>primeramente, sustituyendo $x = \\simplify{-{c}}$&nbsp; sobre $p(x)$ obtenemos&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>pero, por el teorema del resto $p(\\simplify{-{c}}) = \\var{rem1}$ (usando la primera vi&ntilde;eta), entonces esto se convierte en</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>Similarmente, sustituyendo $x = \\simplify{-{d}}$&nbsp;sobre&nbsp; $p(x)$, obtenemos</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>pero, por el teorema del resto $p(\\simplify{-{d}}) = \\var{rem2}$ (usando la segunda vi&ntilde;eta), entonces esto se convierte en</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>Tenemos ahora dos ecuaciones simultaneas para $s$ y $t$:</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></p>\n<p>A continuaci&oacute;n, restamos la segunda ecuaci&oacute;n de la primera ecuaci&oacute;n.</p>\n<p>Esto nos permite cancelar los t&eacute;rminos que involucran $ t $ y nos da una ecuaci&oacute;n solo en t&eacute;rminos de $ s $, que luego podemos reorganizar para encontrar el valor de $ s $.</p>\n<p>Restar las dos ecuaciones da:</p>\n<p></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>Entonces, podemos reorganizar esta ecuaci&oacute;n para que</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>Podemos calcular $ t $ sustituyendo nuestro valor de $ s $ en una de nuestras ecuaciones simult&aacute;neas originales. Por ejemplo, usemos la ecuaci&oacute;n</p>\n<p>\\[\\simplify[all,fractionnumbers]{s*{(-{d})^2}+t} = \\simplify[all,fractionnumbers]{{rem2}-{coef2_x3*(-{d})^3}-{coef2_x*(-{d})}}.\\]</p>\n<p>Sustituyendo nuestro valor de $ s $ en esta ecuaci&oacute;n nos da</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>Esta misma respuesta tambi&eacute;n se hubiera obtenido si hubi&eacute;semos sustituido nuestro valor de $ s $ en la otra ecuaci&oacute;n.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"coef2_x": {"name": "coef2_x", "group": "Ungrouped variables", "definition": "random(-3..3 except 0 except c except d)", "description": "<p>Coefficient of x.</p>", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-3..3 except 0 except 1)", "description": "<p>Dividing term 1.</p>", "templateType": "anything", "can_override": false}, "rem2": {"name": "rem2", "group": "Ungrouped variables", "definition": "random(-3..3 except 0) ", "description": "<p>Second remainder.</p>", "templateType": "anything", "can_override": false}, "numerator": {"name": "numerator", "group": "Ungrouped variables", "definition": "(rem1 - rem2-coef2_x3*(-c)^3-coef2_x*(-c)+coef2_x3*(-d)^3+coef2_x*(-d))*(-d)^2", "description": "<p>Numerator of s</p>", "templateType": "anything", "can_override": false}, "denominator": {"name": "denominator", "group": "Ungrouped variables", "definition": "(-c)^2-(-d)^2", "description": "<p>Denominator of s.</p>", "templateType": "anything", "can_override": false}, "coef2_x3": {"name": "coef2_x3", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "<p>Coefficient of x^3.</p>", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(-(c))^2", "description": "<p>Simplifies&nbsp; first coefficient of s.</p>", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-3..3 except c except -c except 0 except 1)", "description": "<p>Dividing term 2.</p>", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "(-(d))^2", "description": "<p>Simplifies second coefficient of s.</p>", "templateType": "anything", "can_override": false}, "rem1": {"name": "rem1", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "<p>First remainder.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "x > y", "maxRuns": 100}, "ungrouped_variables": ["c", "d", "coef2_x3", "coef2_x", "rem1", "rem2", "numerator", "denominator", "x", "y"], "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>Use el teorema del resto para hallar el resto cuando $p(x) $ se divide por $(\\simplify {x + {c}}) $, cree una ecuaci&oacute;n que incluya $ s $ y $ t $.</p>\n<p><br/>[[0]]$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>El teorema del resto establece que si un polinomio $f(x)$ se divide por $ (\\simplify{a*x-b})$, entonces el resto es $ f (\\frac {b}{a}) $.</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": "{c}^2", "maxValue": "{c}^2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "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": "{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, "displayAnswer": "", "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>Use el teorema del resto para el resto cuando $p(x)$ se divide por $(\\simplify{x+{d}}) $, cree otra ecuaci&oacute;n que incluya $ s $ y $ 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, "displayAnswer": "", "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, "displayAnswer": "", "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>Encuentra el valor de $ s $. Reduce tu respuesta a su forma fraccionaria m&aacute;s simple.</p>\n<p>$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></p>\n<p>Reste las dos ecuaciones simult&aacute;neas para $s$ y $t$, obtenidas en las partes a) y b), una de la otra.Luego reorganice esta nueva ecuaci&oacute;n para encontrar el valor de $s$.</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": "{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}/{(-c)^2-(-d)^2}", "maxValue": "{rem1 - coef2_x3*(-c)^3-coef2_x*(-c)-rem2+coef2_x3*(-d)^3+coef2_x*(-d)}/{(-c)^2-(-d)^2}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": 0, "displayAnswer": "", "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>Encuentra el valor de $t$. Reduce tu respuesta a su forma fraccionaria m&aacute;s simple.</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>Sustituya el valor de $s$ de la parte c) en una de las ecuaciones simult&aacute;neas por $s$ y $t$.</p>\n<p>Luego, reorganiza esta ecuaci&oacute;n para encontrar el valor de $t$.</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, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "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/"}]}