Polynomial Question (Need Help) (1 Viewer)

Andy005 · Apr 13, 2017

Write down an odd monic polynomial of least degree with a triple root of x=-2 and a root of x=1

InteGrand · Apr 13, 2017

Andy005 said:
Write down an odd monic polynomial of least degree with a triple root of x=-2 and a root of x=1

$\noindent Let the desired polynomial be $P(x)$. Since $P$ is odd, we must have that $2$ is also a triple root and $-1$ is also a root. So $P(x) = (x+2)^{3}(x-2)^{3}(x+1)(x-1)Q(x) = \left(x^{2} -4\right)^{3}\left(x^{2} -1\right)Q(x)$ for some polynomial $Q(x)$ (which must be monic for $P(x)$ to be monic). The part without $Q(x)$ now is even, so to make $P(x)$ odd with least degree, take $Q(x) = x$ (can't use a constant polynomial for $Q(x)$ since then $P(x)$ would not be odd).$

Andy005 · Apr 13, 2017

Thanks for the help!

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Polynomial Question (Need Help) (1 Viewer)

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