How do I find multiplicities for functions?
I assume that you are asking for the multiplicities of the roots. For example `f(x)=x^3+x^2-5x+3` factors as `f(x)=(x-1)^2(x+3)` , so f(x) has roots 1 and -3, with 1 as a double root or a root of multiplicity 2.
Generally you factor the polynomial -- the power on the linear factor is the multiplicity. From the graph you know if the multiplicity is even or odd -- with an even multiplicity the graph touches the x-axis without going "through" it. In other words, the function has a local maximum or minimum at that zero. If the multiplicity is odd, the graph goes through the axis.
For the example above:
We wish to find
`` So we can find multiplicity of f as
`` This method we apply