Führen Sie entsprechend der Anleitung die folgende Polynomdivision durch: $f = \\var{f}$ geteilt durch $g = \\var{g}$.
\nMachen Sie sich am besten bei der Bearbeitung der Aufgabe Notizen in Form des üblichen Schemas für die Polynomdivision. Sie bekommen zwischendurch angezeigt, auf welchem Stand Ihre Notizen sein sollten.
\nWenn Sie noch unsicher sind, was gemeint ist, können Sie auch direkt zum Endergebnis springen (und danach, wenn Sie möchten, die Aufgabe mit anderen Polynomen neu beginnen).
", "advice": "Die einzelnen Schritte sind hoffentlich selbsterklärend. Vergleichen Sie jeweils mit den angegebenen \"Zwischenständen\" und mit der vollständigen Polynomdivision am Ende.
\nFragen Sie gegebenenfalls gerne nach!
\nIn diesem Fall ist der Rest $0$. Wir haben also $f = \\var{f}$ als Produkt von $\\var{q}$ und $g = \\var{g}$ geschrieben. Dies kann man nutzen, um die Nullstellen von $f$ zu bestimmen: Für den linearen Teil ist die Nullstelle unmittelbar zu ermitteln. Für den anderen Teil ist nur noch eine quadratische Gleichung zu lösen.
", "rulesets": {}, "extensions": ["polynomials"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"overview_1": {"name": "overview_1", "group": "Ungrouped variables", "definition": "overview(1, overview_l_list)", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "polynomial(x, [random(-2, -1, 1, 2, 3), random(-1, 1, 2, 3)])", "description": "", "templateType": "anything", "can_override": false}, "r1": {"name": "r1", "group": "Ungrouped variables", "definition": "qq2 * g", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "f - r1", "description": "", "templateType": "anything", "can_override": false}, "ff3": {"name": "ff3", "group": "Ungrouped variables", "definition": "f[3] * x^3", "description": "", "templateType": "anything", "can_override": false}, "gg1": {"name": "gg1", "group": "Ungrouped variables", "definition": "g[1] * x", "description": "", "templateType": "anything", "can_override": false}, "qq2": {"name": "qq2", "group": "Ungrouped variables", "definition": "q[2] * x^2", "description": "", "templateType": "anything", "can_override": false}, "qq1": {"name": "qq1", "group": "Ungrouped variables", "definition": "q[1] * x", "description": "", "templateType": "anything", "can_override": false}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "polynomial(x, repeat(random(-2..2), degree(g)))", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "polynomial(x, repeat(random(-2, -1, 1, 2, 3), 3))", "description": "Notiz: Es wäre schön, auch Koeffizienten $=0$ zuzulassen, aber das würde mehr Arbeit verursachen, weil sich die Anzahl der Schritte dann ändert ...
", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "polynomial(x, [0, 1])", "description": "", "templateType": "anything", "can_override": false}, "qq0": {"name": "qq0", "group": "Ungrouped variables", "definition": "q[0]", "description": "", "templateType": "anything", "can_override": false}, "f12": {"name": "f12", "group": "Ungrouped variables", "definition": "f1[2]*x^2", "description": "", "templateType": "anything", "can_override": false}, "overview_pd": {"name": "overview_pd", "group": "Ungrouped variables", "definition": "overview(4, overview_l_list)", "description": "", "templateType": "anything", "can_override": false}, "tt": {"name": "tt", "group": "Ungrouped variables", "definition": "poltoarray(g, 3)", "description": "", "templateType": "anything", "can_override": false}, "r2": {"name": "r2", "group": "Ungrouped variables", "definition": "g*qq1", "description": "", "templateType": "anything", "can_override": false}, "f2": {"name": "f2", "group": "Ungrouped variables", "definition": "f1-r2", "description": "", "templateType": "anything", "can_override": false}, "r3": {"name": "r3", "group": "Ungrouped variables", "definition": "g*qq0", "description": "", "templateType": "anything", "can_override": false}, "f3": {"name": "f3", "group": "Ungrouped variables", "definition": "f2-r3", "description": "", "templateType": "anything", "can_override": false}, "coeff_f_nonzero": {"name": "coeff_f_nonzero", "group": "Ungrouped variables", "definition": "(f[0] * f[1] * f[2] * f[3]) != 0", "description": "", "templateType": "anything", "can_override": false}, "t1": {"name": "t1", "group": "Ungrouped variables", "definition": "0", "description": "", "templateType": "anything", "can_override": false}, "overview_l_list": {"name": "overview_l_list", "group": "Ungrouped variables", "definition": "[\n safe(\"\\\\begin{array}[t]{cc\") +\n join(repeat(\"c\", 2*degree(f)), \"\") +\n \"} & \" +\n poltoarray(f, degree(f)) +\n \" \\\\\\\\ -( & \" +\n poltoarray(r1, degree(f)) +\n \" ) \\\\\\\\\\\\hline\",\n \" & \" +\n poltoarray(f1, degree(f)) +\n \"\\\\\\\\ -( & \" +\n poltoarray(r2, degree(f)) +\n \" ) \\\\\\\\ \\\\hline\",\n \" & \" +\n poltoarray(f2, degree(f)) +\n \"\\\\\\\\ -( & \" +\n poltoarray(r3, degree(f)) +\n \" ) \\\\\\\\ \\\\hline\",\n \" & \" +\n poltoarray(f3, degree(f))\n]", "description": "", "templateType": "anything", "can_override": false}, "overview_2": {"name": "overview_2", "group": "Ungrouped variables", "definition": "overview(2, overview_l_list)", "description": "", "templateType": "anything", "can_override": false}, "overview_3": {"name": "overview_3", "group": "Ungrouped variables", "definition": "overview(3, overview_l_list)", "description": "", "templateType": "anything", "can_override": false}, "f21": {"name": "f21", "group": "Ungrouped variables", "definition": "f2[1]*x", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "q*g+r", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "(f[1] * f[2] * f[3] * f1[1]) != 0", "maxRuns": 100}, "ungrouped_variables": ["g", "r1", "f1", "ff3", "gg1", "qq2", "qq1", "r", "q", "x", "qq0", "f12", "tt", "r2", "f2", "r3", "f3", "coeff_f_nonzero", "t1", "overview_pd", "overview_l_list", "overview_1", "overview_2", "overview_3", "f21", "f"], "variable_groups": [], "functions": {"poltoarray": {"parameters": [["p", "polynomial"], ["sz", "number"]], "type": "anything", "language": "jme", "definition": "join(repeat(\" \", 2*(sz-degree(p))) + map(x -> replace_regex(\"\\\\*\", \" \", x, \"g\"), split(string(p), \" \") ), \" & \")"}, "overview": {"parameters": [["l", "number"], ["oll", "list of string"]], "type": "string", "language": "jme", "definition": "latex(\n safe(\"\\\\begin{array}{ll} \") +\n overview_l(l, oll) +\n \" & \" +\n overview_r(l) +\n safe(\" \\\\end{array}\")\n)"}, "overview_r": {"parameters": [["l", "number"]], "type": "anything", "language": "jme", "definition": "safe(\"\\\\begin{array}[t]{cccc} = & (\") +\nif(l < 3, string_cdot(highest_terms(q, l)) + \"\\\\ +\\\\ ?\", string_cdot(q)) +\n\") \\\\cdot (\" + string_cdot(g) + \") & + & \" +\nif(l < 4,\n \"?\",\n if(r[0] >= 0,\n string(r),\n \"(\" + string(r) + \")\"\n )\n) +\nsafe(\"\\\\end{array}\")"}, "string_cdot": {"parameters": [["p", "polynomial"]], "type": "string", "language": "jme", "definition": "replace_regex(\"\\\\*\", \" \", string(p), \"g\")"}, "overview_l": {"parameters": [["l", "number"], ["oll", "list of string"]], "type": "string", "language": "jme", "definition": "join(oll[0..l], \"\") + safe(\"\\\\end{array}\")"}, "highest_terms": {"parameters": [["p", "polynomial"], ["i", "number"]], "type": "polynomial", "language": "jme", "definition": "// return the polynomial obtained from p by keeping only the i highest summands\nfoldl(\n (t, z) -> t+z,\n polynomial(x, [0]),\n map(j -> p[j] * polynomial(x, [0, 1])^j, (degree(p)+1-i)..degree(p))\n)"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Schritt 1.1", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Schritt 1.2", "rawLabel": "", "otherPart": 1, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "\u00dcbersicht Endergebnis", "rawLabel": "", "otherPart": 10, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Teil 1", "prompt": "Teilen Sie zunächst die Summanden mit höchstem Exponenten von $f$ und $g$ durcheinander:
\n$\\var{ff3}$ geteilt durch $\\var{gg1}$ ist [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expr(qq2)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Schritt 1.2", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Schritt 1.3", "rawLabel": "", "otherPart": 2, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "\u00dcbersicht Endergebnis", "rawLabel": "", "otherPart": 10, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Teil 1", "prompt": "Multiplizieren Sie jetzt das Ergebnis des vorherigen Schritts mit $g$:
\n$\\var{qq2} \\cdot (\\var{g}) =$ [[0]].
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expr(r1)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Schritt 1.3", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Schritt 2.1", "rawLabel": "", "otherPart": 3, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "\u00dcbersicht Endergebnis", "rawLabel": "", "otherPart": 10, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Teil 1", "prompt": "Wir schreiben das bisherige Ergebnis nach dem üblichen Schema auf
\n\\[
\\var{overview_1}
\\]
Subtrahieren Sie nun
\n$\\var{f} - (\\var{r1}) =$ [[0]].
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expr(f1)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Schritt 2.1", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Schritt 2.2", "rawLabel": "", "otherPart": 4, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "\u00dcbersicht Endergebnis", "rawLabel": "", "otherPart": 10, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Teil 2", "prompt": "Der erste Teil der Polynomdivision ist nun erledigt. Die nächsten Schritte sind ganz analog zu den vorherigen, nur dass wir nun das Polynom $\\var{f1}$ (also die Differenz, die wir gerade eben ausgerechnet haben) durch $\\var{g}$ (hier ändert sich nichts) teilen.
\nTeilen Sie zunächst die Leitkoeffizienten durcheinander:
\n$\\var{f12}$ geteilt durch $\\var{gg1}$ ist [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expr(qq1)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Schritt 2.2", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Schritt 2.3", "rawLabel": "", "otherPart": 5, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "\u00dcbersicht Endergebnis", "rawLabel": "", "otherPart": 10, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Teil 2", "prompt": "Multiplizieren Sie das Ergebnis mit $g$.
\n$\\var{qq1}\\cdot (\\var{g}) = $ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expr(qq1*g)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Schritt 2.3", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Schritt 3.1", "rawLabel": "", "otherPart": 6, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "\u00dcbersicht Endergebnis", "rawLabel": "", "otherPart": 10, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Teil 2", "prompt": "Notieren wir zuerst den aktuellen Zwischenstand nach dem üblichen Schema für die Polynomdivision:
\n\\[
\\var{overview_2}
\\]
Führen Sie nun die Subtraktion durch:
\n$\\var{f1} - (\\var{r2}) =$ [[0]].
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expr(f1-qq1*g)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Schritt 3.1", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Schritt 3.2", "rawLabel": "", "otherPart": 7, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "\u00dcbersicht Endergebnis", "rawLabel": "", "otherPart": 10, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Teil 3", "prompt": "Es geht nun weiter mit dem \"verbleibenden Rest\" $\\var{f2}$. Teilen Sie zunächst die Leitkoeffizienten:
\n$\\var{f21}$ geteilt durch $\\var{gg1}$ ist [[0]].
\n", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{qq0}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Schritt 3.2", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Schritt 3.3", "rawLabel": "", "otherPart": 8, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "\u00dcbersicht Endergebnis", "rawLabel": "", "otherPart": 10, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Teil 3", "prompt": "Multiplizieren Sie nun $\\var{qq0} \\cdot (\\var{g}) = $ [[0]].
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expr(qq0*g)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Schritt 3.3", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Schritt 4", "rawLabel": "", "otherPart": 9, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}, {"label": "\u00dcbersicht Endergebnis", "rawLabel": "", "otherPart": 10, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Teil 3", "prompt": "Der Zwischenstand ist nun
\n\\[ \\var{overview_3} \\]
\nund wir sind fast am Ziel.
\nSubtrahieren Sie zum Schluss $\\var{f2} - (\\var{r3}) =$ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expr(r)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Schritt 4", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "\u00dcbersicht Endergebnis", "rawLabel": "", "otherPart": 10, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Teil 4", "prompt": "Der Grad von $\\var{r}$ ist kleiner als der Grad von $\\var{g}$. Das Verfahren ist damit beendet, und $\\var{r}$ ist der Rest bei der Polynomdivision.
\nInsgesamt erhalten wir also
\n$\\var{f} = ($ [[0]] $)(\\var{g}) + $ [[1]].
\n(Wenn Sie sich keine Notizen gemacht haben, müssen Sie eventuell noch einmal in die vorherigen Schritte zurückspringen, um die Informationen für das erste Kästchen zusammenzusammeln.)
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