Question

Suppose that a polynomial has zeros of 3 with a multiplicity of 2, a zero with a -7 with a multiplicity of 5 and a zero of 4 with a multiplicity of 1. Write a possible function that would have these zeros with multiplicity

Answer 1

Answer:

  f(x) = (x -3)^2·(x +7)^5·(x -4)

Step-by-step explanation:

When a polynomial (or any function) has a zero of "a" with multiplicity "b", then (x -a)^b is a factor. Your polynomial factors as ...

  f(x) = (x -3)^2·(x +7)^5·(x -4)

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Comment on the graph

When the zero has multiplicity 1, the function crosses the x-axis with a non-zero slope.

When a zero has even multiplicity, the graph touches the axis, but does not cross. The graph crosses the x-axis if the multiplicity of the zero is odd.

When the zero has multiplicity greater than 1, the point where the curve touches the x-axis has zero slope. The "flatness" at that point increases with increasing multiplicity.