// Numbas version: exam_results_page_options
{"name": "El teorema del factor establece que si $ f (x) $ es un polinomio y $ f (p) = 0 $, entonces $ (xp) $ es un factor de $ f (x) $.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "El teorema del factor establece que si $ f (x) $ es un polinomio y $ f (p) = 0 $, entonces $ (xp) $ es un factor de $ f (x) $.", "tags": [], "metadata": {"description": "<p>Aplique el teorema del factor para verificar cu&aacute;les de una lista de polinomios lineales son factores de otro polinomio.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><span style=\"vertical-align: inherit;\"><span style=\"vertical-align: inherit;\">El Teorema del Factor establece que si $ f (x) $ es un polinomio y $ f (p) = 0 $, entonces $ (x-p) $ es un factor de $ f (x) $.</span></span></p>\n<p></p>", "advice": "<p>Para encontrar los factores del polinomio $f(x) = \\simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}$,&nbsp;usamos el teorema del factor.</p>\n<p>Si $f(x)$ es un polinomio y $f(p) = 0$, entonces $(x-p)$ es un factor de $f(x)$.</p>\n<p>Si $(\\simplify{(x+{a})})$ es un factor de $f(x)$ entonces por el teorema del factor,&nbsp;$f(\\simplify{-{a}}) = 0$.</p>\n<p>Vemos que:\\[<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>Por lo tanto, $(\\simplify{(x+{a})})$ es un factor de $f(x)$. Similarmente para $(\\simplify{(x+{d})})$,</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>Por lo tanto, $(\\simplify{(x+{d})})$ no es un factor de $f(x)$.&nbsp;finalmente, para $(\\simplify{(x+{c})})$,</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>Por lo tanto, $(\\simplify{(x+{c})})$ es tambi&eacute;n un factor de $f(x)$.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"coef2_x2": {"name": "coef2_x2", "group": "Ungrouped variables", "definition": "(a+b+c)*(-d)^2", "description": "<p>Number obtained from putting x=-d into the second term of the equation.</p>", "templateType": "anything", "can_override": false}, "coef3_x3": {"name": "coef3_x3", "group": "Ungrouped variables", "definition": "(-c)^3", "description": "<p>Number obtained for putting x=-c into the first term of the equation.</p>", "templateType": "anything", "can_override": false}, "const": {"name": "const", "group": "Ungrouped variables", "definition": "a*b*c", "description": "<p>Constant term in the equation.</p>", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-2..3 except 0 except c)", "description": "<p>N&uacute;mero aleatorio entre -2 y 3, sin incluir 0 para crear polinomio.</p>", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-2..3 except 0)", "description": "<p>Random number between -2 and 3 except 0 for creating polynomial.</p>", "templateType": "anything", "can_override": false}, "coef3_x2": {"name": "coef3_x2", "group": "Ungrouped variables", "definition": "(a+b+c)*(-c)^2", "description": "", "templateType": "anything", "can_override": false}, "coef2_x": {"name": "coef2_x", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-d)", "description": "<p>Number obtained from putting x=-d into the 3rd term for the equation.</p>", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-2..3 except 0 except c)", "description": "<p>Random number between -2 and 3 except 0 for creating polynomial.</p>", "templateType": "anything", "can_override": false}, "coef1_x2": {"name": "coef1_x2", "group": "Ungrouped variables", "definition": "(a+b+c)*(-a)^2", "description": "<p>Number obtained from putting x=-a into the second term of the equation.</p>", "templateType": "anything", "can_override": false}, "coef1_x3": {"name": "coef1_x3", "group": "Ungrouped variables", "definition": "(-a)^3", "description": "<p>Number obtained from putting x=-a into the first term of the equation.</p>", "templateType": "anything", "can_override": false}, "coef2_x3": {"name": "coef2_x3", "group": "Ungrouped variables", "definition": "(-d)^3", "description": "<p>Number obtained from putting x=-d into the&nbsp;first term&nbsp;in the&nbsp;equation.</p>", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-2..2 except 0 except a except c except b)", "description": "<p>Incorrect answer for part a.</p>", "templateType": "anything", "can_override": false}, "coef1_x": {"name": "coef1_x", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-a)", "description": "<p>Number obtained from putting x=-a into the first term of the equation.</p>", "templateType": "anything", "can_override": false}, "coef3_x": {"name": "coef3_x", "group": "Ungrouped variables", "definition": "(a*b+b*c+a*c)*(-c)", "description": "<p>Number obtained by putting x=-c into the third term of the equation.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "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"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "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>Establecer cu&aacute;l o cu&aacute;les de los siguientes son factores del polinomio $f(x) = \\simplify{x^3+({a}+{b}+{c})x^2+({a}{b}+{a}{c}+{b}{c})x+{a}{b}{c}}.$</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["<p>$(\\simplify{x+{a}})$</p>", "<p>$(\\simplify{x+{d}})$</p>", "<p>$(\\simplify{x+{c}})$</p>"], "matrix": ["1", 0, "1"], "distractors": ["", "", ""]}], "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/"}]}