2022-07-16

What is a good technique for solving polynomials?
Say for example:
$6{x}^{3}-17{x}^{2}-4x+3=0$

tiltat9h

Expert

First, to minimize the lead coefficient, we reverse the polynomial (which reciprocates the roots) and then we scale it to make the leading coefficient 1, i.e. we apply the AC-method, which yields

Note $\phantom{\rule{mediummathspace}{0ex}}\mathrm{z}=1\phantom{\rule{mediummathspace}{0ex}}$ is a root. By Vieta, other roots have product $-54/1,\phantom{\rule{thinmathspace}{0ex}}$ sum $4\phantom{\rule{negativethinmathspace}{0ex}}-\phantom{\rule{negativethinmathspace}{0ex}}1=3,\phantom{\rule{mediummathspace}{0ex}}$ so are $\phantom{\rule{mediummathspace}{0ex}}9,\phantom{\rule{thinmathspace}{0ex}}-6.\phantom{\rule{mediummathspace}{0ex}}$
Hence $\phantom{\rule{mediummathspace}{0ex}}\mathrm{z}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}1,\phantom{\rule{thinmathspace}{0ex}}9,\phantom{\rule{thinmathspace}{0ex}}-6\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}$ so $\phantom{\rule{mediummathspace}{0ex}}\mathrm{x}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\mathrm{z}/3\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}1/3,\phantom{\rule{thinmathspace}{0ex}}3,\phantom{\rule{thinmathspace}{0ex}}-2,\phantom{\rule{thinmathspace}{0ex}}$ which reciprocated yields $\phantom{\rule{mediummathspace}{0ex}}3,\phantom{\rule{thinmathspace}{0ex}}1/3,\phantom{\rule{thinmathspace}{0ex}}-1/2.$

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