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If I have $$x^2(x-3)(x+3)=0$$ then the solutions are: $$x_{1,2}=0, x_3=3, x_4=-3 $$ or $$x_{1}=0, x_2=3, x_3=-3?$$ So are there 4 or 3 solutions?

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There are $3$ distinct solutions but there are $4$ roots. This has to do with the multiplicity of the roots in a polynomial. It is often times more in vogue to say that there are $3$ solutions to this but, technically, it must be said that there are $3$ distinct roots, as the Fundamental Theorem of Algebra says that there are $d$ roots in $\mathbb{C}$ for a polynomial of degree $d$. Here, $d=4$.

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  • $\begingroup$ So 0 is an even zero of this function and -3 and 3 are odd zeros? Is that correct? $\endgroup$ – L_McClain Dec 1 '13 at 15:53
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    $\begingroup$ @user111410 Yes, if parity matters for you. $\endgroup$ – Ahaan S. Rungta Dec 1 '13 at 15:55

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