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:

Let f(x)=ax^2+bx+c

We wish to find

`f^n(x)`

`` So we can find multiplicity of f as

`f^2(x)=f(x)*f(x)`

`f^3(x)=f(x)*f^2(x)`

`.`

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`f^n(x)=f(x)*f^(n-1)(x)`

`` This method we apply

`f^2(x)=(ax^2+bx+c)*(ax^2+bx+c)`

`=a^2x^4+abx^3+acx^2+abx^3+b^2x^2+bcx+acx^2+cbx+c^2`

`=a^2x^4+2abx^3+2acx^2+2bcx+b^2x^2+c^2`