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多项式合成除法练习

多项式合成除法练习
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2025-09-06 19:26
函数函数图像
所属专题 Arithmetic and Algebra
本范例演示多项式合成除法的完整计算过程，帮助学习者掌握这一重要代数技巧。核心数学概念包括多项式除法、合成除法（综合除法）、余数定理与因式定理。通过随机生成不同系数的多项式，让用户逐步输入除数、商和余数，系统自动验证答案正确性。构造要点：利用Polynomial命令构造多项式，用Div命令实现除法运算，Coefficients提取系数，If和KeepIf进行条件判断与筛选，Textfield实现交互输入，Shuffle实现随机题目生成。适合中学高年级及大学预科数学课程，用于巩固多项式运算技能。

GeoGebra 命令及参数

  1. Shuffle({1, 2, 3}) → allCaseNum, {1, 2, 3}
  2. Shuffle(Sequence[t, t, -5, 5, 1] \ {0}) → allA, {-4, 1, -3, 5, 2, -1, -5, -2, 3, 4}
  3. Shuffle(Sequence[t, t, -5, 5, 1] \ {0}) → allB, {-4, 3, 4, -3, 1, -1, 2, 5, -2, -5}
  4. Shuffle(Join[Sequence[Sequence[(j, k), j, -4, 4, 1], k, -4, 4, 1]]) → startGrid, {(4, 3), (-1, -1), (-2, -3), (-2, 0), (3, 2), (0, -3), (0, -2), (4, -3), (-3, 0), (1, -3), (-3, -3), (-1, -3), (1, -1), (-3, 2), (-4, 2), (4, -2), (-2, -1), (4, 4), (0, -4), (-1, 2), (4, 2), (-1, -2), (-3, 4), (1, 3), (-2, 1), (2, 2), (3, -1), (-4, -3), (3, -2), (-2, 4), (-1, 4), (-2, -2), (0, 4), (3, 3), (-1, 1), (-2, 3), (0, -1), (2, 4), (-3, -1), (-1, 0), (0, 0), (-4, -1), (0, 3), (-4, 0), (2, 3), (2, -2), (-4, 1), (-4, -2), (-4, 3), (-4, -4), (-3, 1), (-2, -4), (2, -3), (1, 1), (-1, -4), (2, -4), (1, -4), (3, 0), (4, -1), (3, -4), (-1, 3), (-3, -4), (-4, 4), (-2, 2), (2, -1), (1, 0), (3, 4), (4, 0), (2, 0), (2, 1), (1, -2), (4, -4), (1, 4), (-3, 3), (1, 2), (3, -3), (3, 1), (0, 1), (-3, -2), (4, 1), (0, 2)}
  5. Element(allCaseNum, Mod[countQuestion, Length[allCaseNum]] + 1) → caseNum
  6. Element(allA, Mod[countQuestion, Length[allA]] + 1) → valA
  7. Element(allB, Mod[countQuestion, Length[allB]] + 1) → valB
  8. Element(startGrid, Mod[countQuestion, Length[startGrid]] + 1) → V1
  9. Shuffle(Sequence[t, t, -5, 5, 1] \ {0}) → allC, {-5, 1, -3, 5, -1, 3, 4, 2, -2, -4}
  10. Shuffle(Sequence[t, t, -5, 5, 1] \ {0}) → allD, {5, 1, 2, -5, -1, 4, -3, -2, -4, 3}
  11. Shuffle(Sequence[t, t, -5, 5, 1] \ {0}) → allE, {-4, -5, -2, -1, 3, 4, 1, 2, -3, 5}
  12. Element(allC, Mod[countQuestion, Length[allC]] + 1) → valC
  13. Element(allD, Mod[countQuestion, Length[allD]] + 1) → valD
  14. Element(allE, Mod[countQuestion, Length[allE]] + 1) → valE
  15. Polynomial(If[caseNum ≟ 1, (((x - valA) * (x - valB)) * (x - valC)), caseNum ≟ 2, ((((x - valA) * (x - valB)) * (x - valC)) * (x - valD)), (((((x - valA) * (x - valB)) * (x - valC)) * (x - valD)) * (x - valE))]) → givenPoly
  16. KeepIf(x ≤ 6, DivisorsList[abs(tn)]) → allDivis
  17. Shuffle({-1, 1}) → allPM, {1, -1}
  18. Element(allPM, Mod[countQuestion, Length[allPM]] + 1) → valPM
  19. Element(allDivis, Mod[countQuestion, Length[allDivis]] + 1) → divis
  20. Polynomial(x + (valPM * divis)) → givenDivis
  21. Div(givenPoly, givenDivis) → givenQuot
  22. Coefficients(givenPoly) → allCoeff
  23. Mod(givenPoly, givenDivis) → givenRem
  24. If(caseNum ≟ 1, 7, caseNum ≟ 2, 9, caseNum ≟ 3, 11) → narrativeChange
  25. Textfield(inputB) → InputBoxB
  26. Corner(5) → layoutAppSize
  27. onboardingTextOffset(Corner[onboardingText, 1], Corner[onboardingText, 3], onboardingCase, Corner[1], Corner[3], Corner[5]) → OnboardingTextOffset
  28. Textfield(inputA) → InputBoxA
  29. Textfield(inputC) → InputBoxC
  30. Polygon(P1, Mirror[P1, yAxis], Mirror[P1, (0, 0)], Mirror[P1, xAxis]) → peep, p1, f, g, h
  31. Shuffle(Sequence[t, t, -6, 6, 1] \ {0, -1, 1}) → allH, {3, -3, -2, 6, -4, 5, 2, 4, -5, -6}
  32. If(caseNum ≟ 1, 9, caseNum ≟ 2, 11, 13) → aniMax

AI 绘图提示语参考

请生成一个用于多项式综合除法练习的代数图形：先在坐标平面上给出若干随机选取的多项式根位置点，它们来自若干从 -5 到 5（不含 0）的整数集合，并根据选中的情形生成给定多项式为若干 (x−val) 因子的乘积（包含从 3 次到 5 次的不同情形）。同时在同一画面中构造一个与之对应的除式，多项式除式应为一次多项式 x 加上一个由整数点乘得到的常数项；接着再呈现综合除法的结果：用同一组多项式表达式给出商多项式并计算余数（余数为给定多项式除以除式后的结果）。整体包含坐标网格与用于输入/对比的区域，让学习者按步骤观察给定多项式、除式、商与余数之间的关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。