# Analytic Geometry: Trace the curve of x^{3}-6x^{2}+11x-y=6

Douglas Kraatz 2021-11-27 Answered
Analytic Geometry:
Trace the curve of $$\displaystyle{x}^{{{3}}}-{6}{x}^{{{2}}}+{11}{x}-{y}={6}$$

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$$\displaystyle{x}^{{{3}}}-{6}{x}^{{{2}}}+{11}{x}-{y}={6}$$
$$\displaystyle{y}={x}^{{{3}}}-{6}{x}^{{{2}}}+{11}{x}-{6}$$
$$\displaystyle{P}{\left({x}\right)}={x}^{{{3}}}-{6}{x}^{{{2}}}+{11}{x}-{6}$$
$$\displaystyle{P}{\left({z}\right)}={1}-{6}+{11}-{6}={0}$$
(x-l) is a talov of p(x)
Diviwng p(x) by (x-l) we get
$$\displaystyle{2}{\left({x}\right)}={x}^{{{2}}}-{3}{x}-{2}{x}+{6}$$
$$\displaystyle{2}{\left({x}\right)}={x}{\left({x}-{3}\right)}-{2}{\left({x}-{3}\right)}$$
$$\displaystyle{2}{\left({x}\right)}={\left({x}-{2}\right)}{\left({x}-{3}\right)}$$
$$\displaystyle{p}{\left({x}\right)}={\left({x}-{1}\right)}{2}{\left({x}\right)}$$
$$\displaystyle{p}{\left({x}\right)}={\left({x}-{1}\right)}{\left({x}-{2}\right)}{\left({x}-{3}\right)}$$
Zowes of polynaval are x=1,2,3