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三项式展开法

三项式展开法
几何画板及绘图内容加载中，请稍等……
2025-01-09 19:34
微积分导数
所属专题 Binomische Formeln
本范例演示了如何使用三项式展开法来计算二项式的幂。用户可以输入任意的二项式表达式，系统会自动展开并计算结果。这个范例涉及的核心数学概念是三项式展开法和二项式的幂计算。作图要点包括使用 GeoGebra 的 Expand 命令来展开二项式表达式，使用 Text 命令来显示结果。适合的学段是初中和高中数学，特别是代数和数论。

GeoGebra 命令及参数

  1. Expand(TaskFunc) → solFunc
  2. Text(inputFunc) → txtInput
  3. Sequence(Shuffle[1…3], k, 1, 100) → OrderList1, {{3, 2, 1}, {3, 1, 2}, {3, 2, 1}, {2, 3, 1}, {3, 2, 1}, {3, 1, 2}, {2, 1, 3}, {2, 1, 3}, {2, 3, 1}, {1, 2, 3}, {2, 3, 1}, {3, 2, 1}, {1, 3, 2}, {3, 1, 2}, {3, 1, 2}, {3, 1, 2}, {3, 1, 2}, {3, 2, 1}, {2, 3, 1}, {1, 3, 2}, {2, 3, 1}, {2, 1, 3}, {1, 2, 3}, {2, 3, 1}, {2, 3, 1}, {3, 1, 2}, {2, 1, 3}, {3, 1, 2}, {1, 2, 3}, {1, 2, 3}, {3, 1, 2}, {2, 3, 1}, {2, 3, 1}, {2, 3, 1}, {1, 3, 2}, {2, 3, 1}, {3, 2, 1}, {1, 2, 3}, {1, 3, 2}, {3, 1, 2}, {3, 2, 1}, {2, 1, 3}, {1, 3, 2}, {1, 2, 3}, {1, 3, 2}, {2, 3, 1}, {3, 2, 1}, {3, 2, 1}, {3, 1, 2}, {2, 3, 1}, {2, 3, 1}, {1, 3, 2}, {2, 1, 3}, {3, 1, 2}, {2, 1, 3}, {3, 2, 1}, {2, 1, 3}, {2, 3, 1}, {1, 3, 2}, {2, 1, 3}, {1, 2, 3}, {2, 3, 1}, {2, 1, 3}, {1, 3, 2}, {2, 1, 3}, {2, 1, 3}, {1, 3, 2}, {2, 3, 1}, {2, 1, 3}, {3, 1, 2}, {2, 3, 1}, {3, 1, 2}, {2, 3, 1}, {2, 3, 1}, {2, 3, 1}, {1, 3, 2}, {3, 2, 1}, {1, 3, 2}, {1, 3, 2}, {2, 3, 1}, {3, 2, 1}, {1, 3, 2}, {3, 1, 2}, {3, 1, 2}, {1, 2, 3}, {2, 1, 3}, {2, 1, 3}, {3, 1, 2}, {3, 2, 1}, {3, 2, 1}, {3, 1, 2}, {3, 1, 2}, {2, 1, 3}, {3, 2, 1}, {3, 1, 2}, {3, 2, 1}, {2, 3, 1}, {3, 2, 1}, {2, 3, 1}, {1, 2, 3}}
  4. Sequence(Shuffle[1…3], k, 1, 100) → OrderList2, {{3, 1, 2}, {1, 3, 2}, {3, 1, 2}, {1, 2, 3}, {2, 1, 3}, {2, 3, 1}, {3, 2, 1}, {3, 1, 2}, {1, 3, 2}, {3, 2, 1}, {3, 1, 2}, {1, 2, 3}, {3, 2, 1}, {3, 2, 1}, {2, 1, 3}, {1, 3, 2}, {1, 3, 2}, {3, 1, 2}, {3, 2, 1}, {2, 1, 3}, {2, 1, 3}, {1, 3, 2}, {1, 3, 2}, {2, 3, 1}, {3, 2, 1}, {1, 2, 3}, {1, 2, 3}, {2, 3, 1}, {2, 3, 1}, {2, 1, 3}, {2, 1, 3}, {3, 1, 2}, {1, 3, 2}, {3, 1, 2}, {2, 3, 1}, {2, 3, 1}, {3, 2, 1}, {1, 2, 3}, {2, 1, 3}, {1, 3, 2}, {3, 1, 2}, {3, 2, 1}, {1, 2, 3}, {1, 3, 2}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {1, 2, 3}, {2, 3, 1}, {2, 1, 3}, {1, 2, 3}, {2, 3, 1}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3}, {2, 1, 3}, {2, 3, 1}, {1, 2, 3}, {3, 2, 1}, {3, 1, 2}, {1, 3, 2}, {2, 1, 3}, {2, 1, 3}, {2, 1, 3}, {2, 1, 3}, {1, 2, 3}, {1, 3, 2}, {2, 3, 1}, {3, 2, 1}, {3, 2, 1}, {1, 2, 3}, {1, 3, 2}, {1, 2, 3}, {3, 2, 1}, {2, 3, 1}, {2, 3, 1}, {1, 3, 2}, {1, 2, 3}, {3, 2, 1}, {2, 1, 3}, {3, 1, 2}, {3, 1, 2}, {1, 2, 3}, {3, 2, 1}, {3, 2, 1}, {3, 2, 1}, {1, 3, 2}, {3, 2, 1}, {1, 2, 3}, {3, 2, 1}, {3, 1, 2}, {3, 1, 2}, {1, 2, 3}, {1, 2, 3}, {3, 2, 1}, {2, 1, 3}, {1, 2, 3}, {3, 2, 1}, {3, 1, 2}}
  5. Sequence(Shuffle[1…3], k, 1, 100) → OrderList3, {{3, 1, 2}, {2, 1, 3}, {2, 1, 3}, {3, 1, 2}, {2, 1, 3}, {3, 1, 2}, {2, 3, 1}, {1, 2, 3}, {2, 1, 3}, {3, 1, 2}, {3, 2, 1}, {3, 1, 2}, {3, 1, 2}, {3, 1, 2}, {3, 2, 1}, {3, 1, 2}, {1, 2, 3}, {3, 2, 1}, {2, 1, 3}, {1, 3, 2}, {1, 2, 3}, {2, 3, 1}, {3, 2, 1}, {1, 3, 2}, {3, 2, 1}, {2, 3, 1}, {3, 1, 2}, {1, 2, 3}, {1, 2, 3}, {1, 3, 2}, {3, 1, 2}, {3, 1, 2}, {2, 1, 3}, {3, 1, 2}, {1, 2, 3}, {3, 2, 1}, {3, 2, 1}, {3, 2, 1}, {1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, {1, 3, 2}, {2, 3, 1}, {3, 1, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, {2, 3, 1}, {3, 2, 1}, {3, 1, 2}, {1, 2, 3}, {1, 2, 3}, {3, 1, 2}, {1, 2, 3}, {3, 1, 2}, {1, 2, 3}, {1, 2, 3}, {2, 3, 1}, {3, 2, 1}, {2, 1, 3}, {1, 3, 2}, {1, 2, 3}, {3, 1, 2}, {2, 1, 3}, {2, 3, 1}, {1, 3, 2}, {3, 2, 1}, {2, 3, 1}, {1, 3, 2}, {1, 3, 2}, {3, 1, 2}, {2, 1, 3}, {1, 2, 3}, {1, 3, 2}, {1, 3, 2}, {1, 3, 2}, {2, 3, 1}, {3, 2, 1}, {1, 3, 2}, {2, 1, 3}, {2, 1, 3}, {2, 1, 3}, {2, 1, 3}, {2, 1, 3}, {3, 1, 2}, {2, 3, 1}, {1, 2, 3}, {1, 3, 2}, {1, 3, 2}, {3, 2, 1}, {1, 2, 3}, {3, 2, 1}, {2, 3, 1}, {1, 2, 3}, {3, 2, 1}, {1, 2, 3}, {1, 3, 2}, {2, 3, 1}}
  6. Join({{"$\textsf{\phantom{144\:b² }\vphantom{gb²}}$"}, {(("$\textsf{" * DDList1(Order1(1))) * " }\vphantom{gb²}$"), (("$\textsf{" * DDList1(Order1(2))) * " }\vphantom{gb²}$"), (("$\textsf{" * DDList1(Order1(3))) * "\vphantom{gb²}}$")}}) → dropdown1
  7. Join({{"$\textsf{\phantom{144\:b² }\vphantom{gb²}}$"}, {(("$\textsf{" * DDList3(Order3(1))) * " }\vphantom{gb²}$"), (("$\textsf{" * DDList3(Order3(2))) * " }\vphantom{gb²}$"), (("$\textsf{" * DDList3(Order3(3))) * "\vphantom{gb²}}$")}}) → dropdown3
  8. Join({{"$\textsf{\phantom{144\:b² }\vphantom{gb²}}$"}, {(("$\textsf{" * DDList2(Order2(1))) * " }\vphantom{gb²}$"), (("$\textsf{" * DDList2(Order2(2))) * " }\vphantom{gb²}$"), (("$\textsf{" * DDList2(Order2(3))) * "\vphantom{gb²}}$")}}) → dropdown2
  9. SelectedIndex(dropdown1) → indexdd1
  10. SelectedIndex(dropdown2) → indexdd2
  11. SelectedIndex(dropdown3) → indexdd3
  12. SelectedIndex(dropdownSign1) → indexddS1
  13. SelectedIndex(dropdownSign2) → indexddS2
  14. If(Case ≠ 3, indexdd2 - 1 ≟ IndexOf[1, Order2], true) → correctdd2
  15. If(Case ≠ 3, indexddS2 ≟ 2, true) → correctddS2
  16. If(step ≟ 1, correctExtend ∧ correctValue, correctdd1 ∧ correctdd2 ∧ correctdd3 ∧ correctddS1 ∧ correctddS2) → correct
  17. Sequence(Random[1, 10], i, 1, 10) → randomNumbers, {6, 5, 3, 9, 2, 10, 4, 1, 8, 2}
  18. Element(randomNumbers, Mod[questionCount, Length[randomNumbers]] + 1) → randomNumber
  19. If(step ≟ 1, Defined[inputFunc], indexdd1 ≠ 1 ∧ If[Case ≟ 3, true, indexdd2 ≠ 1] ∧ indexdd3 ≠ 1 ∧ indexddS1 ≠ 1 ∧ If[Case ≟ 3, true, indexddS2 ≠ 1]) → clickableBtnValidate
  20. Textfield(inputFunc) → inputBox
  21. If(questionCount ∈ {1, 3, 5, 6, 8, 11}, SignList(1), SignList(2)) → sign1
  22. If(questionCount ∈ {1, 3, 5, 6, 8, 11}, SignList(2), SignList(1)) → sign2
  23. If(CaseVar ≟ 2, "", var1) → txtVar1
  24. If(CaseVar ≟ 1, "", var2) → txtVar2
  25. Invert(pixMat) → invPixMat
  26. If(Case ≟ 3, PosDDS1 + (1.42, 0), PosDDS2 + (1.42, 0)) → PosDD3
  27. ReplaceAll(Text[Expand[TaskFunc]], "-", "+") → txtHint3

AI 绘图提示语参考

需要绘制一个交互式数学计算界面，用于演示立方二项式展开。界面包含一个输入框，用户可输入任意二项式表达式；一个文本显示区域，自动显示展开后的结果；三个下拉菜单，分别列出不同的排列顺序选项；一个验证按钮，用于检查输入是否正确；以及一些文本标签提示操作。构造顺序为：先创建输入框和文本显示区域，然后生成排列列表并填充下拉菜单，接着添加验证按钮，最后调整布局和标签。
以上提示语为根据原始绘图命令整理而来，仅供参考。