# How do you write a polynomial function in standard form with Zero: 0, multiplicity 2?

May 11, 2016

$f \left(x\right) = {x}^{2}$ is one function that meets the specified requirements.

#### Explanation:

If $f \left(x\right)$ has a multiplicity of 2 then for every value in the range for $f \left(x\right)$ there should be 2 solutions.

Note that this would be true for $f \left(x\right) = {x}^{2}$ since if $a$ is a value in the range for $f \left(x\right)$ then there are 2 solutions for $x$, namely $x = - \sqrt{a}$ and $x = + \sqrt{a}$

Note that if $f \left(x\right)$ has a zero at $x = 0$
then $f \left(0\right) = 0$
and again this will be true for $f \left(x\right) = {x}^{2}$

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$f \left(x\right) = {x}^{2}$ is not the only possible function meeting the requirements,
in fact
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = k \cdot {x}^{2}$ for any integer value $k$ will also work.