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Context

If I have an equation which factors to $(x-2)(x-2)(x+3)$, the zeros are 2, 2, and -3. The multiplicity of the zero "2" is 2 because it occurs twice. In graphing this equation, the multiplicity is visually represented by the graph either crossing or "bouncing off" of the x-axis (determined by whether or not the multiplicity is even or odd), and also by the length of the part of the graph with a generally lower-in-value slope.

Question

How can I physically and accurately determine exactly how many times the given equation "spits out" the zero at the given X value, if I am using a common two-dimensional graph? Is there another graphing format that can help me achieve this goal?

Motive

I am asking because I would like to find a way to physically represent an accurate value of multiplicity for an equation, as any lack of representation of any value (using a 2-dimensional graph) can lead to inaccuracy when applying Mathematics to the physical world.

Key Flex
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MJMarquez
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1 Answers1

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The quickest example here is something like $x^2=x \cdot x$. Clearly this function has $0$ as a root of multiplicity 2. Yet, if we look at the graph we see that it only passes through the point $(0,0)$ once. Multiplicty is more of a mathematical concept than a physical one. Graphically, we would only see the distinct roots, not their multiplicty. Algebraically, we look at the factorization which has deeper properties having to do with ring/field theory.

RhythmInk
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