MathPlotAI Logo MathPlotAI

Zeros of Polynomial Functions

Zeros of Polynomial Functions
Loading the math board and drawing, please wait…
2022-06-25 22:57
FunctionCoordinate GeometryOptimizationGraphing
This example demonstrates the distribution and properties of zeros of polynomial functions. Core math concepts include functions, polynomial functions, zeros, median line, tree diagrams, pyramid, difference and slope, quadrilaterals. Construction highlights the use of GeoGebra commands and objects like Sequence, SelectedElement, Point, If, Extremum, Min, Max. Suitable for high school mathematics students and teachers to explore and understand the graphs and properties of polynomial functions.

GeoGebra Commands & Parameters

  1. Sequence(u, u, 2, 7) → nz
  2. SelectedElement(nz) → a
  3. Point(xAxis) → A
  4. Point(xAxis) → B
  5. Point(xAxis) → C
  6. Point(xAxis) → D
  7. Point(xAxis) → E
  8. Point(xAxis) → F
  9. Point(xAxis) → G
  10. If(a ≟ 7, (((((((x - x(A)) * (x - x(B))) * (x - x(C))) * (x - x(D))) * (x - x(E))) * (x - x(F))) * (x - x(G))), If[a ≟ 6, ((((((x - x(A)) * (x - x(B))) * (x - x(C))) * (x - x(D))) * (x - x(E))) * (x - x(F))), If[a ≟ 5, (((((x - x(A)) * (x - x(B))) * (x - x(C))) * (x - x(D))) * (x - x(E))), If[a ≟ 4, ((((x - x(A)) * (x - x(B))) * (x - x(C))) * (x - x(D))), If[a ≟ 3, (((x - x(A)) * (x - x(B))) * (x - x(C))), ((x - x(A)) * (x - x(B)))]]]]) → f
  11. Extremum(f) → H, I, J, K, L, M
  12. Min(y(RemoveUndefined[{H, I, J, K, L, M}])) → b
  13. Max(y(RemoveUndefined[{H, I, J, K, L, M}])) → c
  14. Max(abs(b), abs(c)) → d

AI Drawing Prompt Reference

On the same coordinate plane, plot a polynomial function built from a discrete set of zeros. First generate a sequence of candidate integers, select one value to determine the actual zero set, and place several points on the x-axis at those selected zero locations. Construct the polynomial as the product of (x − x_i) over the chosen x-axis points so the graph crosses the x-axis at each zero. Then find the extremum points of the resulting function, identify the minimum and maximum y-values among the defined extrema, and use the larger of their absolute values as a key amplitude reference.
The prompt above was compiled from the original drawing commands and is for reference only.