 # Let p(x) be a nonzero polynomial of degree less than 1992 having no nonconstant factor in common with x^3 for polynomials f (x) and g(x). necessaryh 2021-09-05 Answered

Let p(x) be a nonzero polynomial of degree less than 1992 having no nonconstant factor in common with ${x}^{3}$ for polynomials f (x) and g(x). Find the smallest possible degree of f (x)?
$\frac{{d}^{1992}}{{dx}^{1992}}\left(\frac{p\left(x\right)}{{x}^{3}-x}\right)=\frac{f\left(x\right)}{g\left(x\right)}$

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Given p(x) be a nonzero polynomial of degree less than 1992 having no non constant factor in common with ${x}^{3}-x$
Let $\frac{{d}^{1992}}{{dx}^{1992}}\left(\frac{p\left(x\right)}{{x}^{3}-x}\right)=\frac{f\left(x\right)}{g\left(x\right)}$, where $f\left(x\right)$ and $g\left(x\right)$ are polynomials.
To find the smallest possible degree of $f\left(x\right)$:
BY division algorithm we know that,
If $f\left(x\right)$ and $g\left(x\right)$ are any two polynomials with $g\left(x\right)\ne 0$, then $f\left(x\right)=g\left(x\right).q\left(x\right)+r\left(x\right)$, where $r\left(x\right)=0$ or deg $r\left(x\right)$ where $q\left(x\right)$ is the quotient and $r\left(x\right)$ is the remainder.
Hence p(x) can be written as
$p\left(x\right)=\left({x}^{3}-x\right)q\left(x\right)+r\left(x\right)$
Here, we can see that the degree of $r\left(x\right)$ is 2 and the degree of $q\left(x\right)$is less than 1989
Then,
$\frac{{d}^{1992}}{{dx}^{1992}}\left(\frac{p\left(x\right)}{{x}^{3}-x}\right)=\frac{{d}^{1992}}{{dx}^{1992}}\left(\frac{r\left(x\right)}{{x}^{3}-x}\right)$
To find $\frac{r\left(x\right)}{{x}^{3}-x}$
Using partial fractions, we have
$\frac{r\left(x\right)}{{x}^{3}-x}=\frac{A}{x-1}+\frac{B}{x}+\frac{C}{x+1}$
Hence,
$\frac{{d}^{1992}}{{dx}^{1992}}\left(\frac{r\left(x\right)}{{x}^{3}-x}\right)=1992!\left(\frac{A}{{\left(x-1\right)}^{1993}}+\frac{B}{{x}^{1993}}+\frac{C}{{\left(x+1\right)}^{1993}}$
$=1992!\left(\frac{A{x}^{1993}{\left(x+1\right)}^{1993}+B{\left(x-1\right)}^{1993}{\left(x+1\right)}^{1993}+C{\left(x-1\right)}^{1993}{x}^{1993}}{{\left(x-1\right)}^{1993}{x}^{1993}{\left(x+1\right)}^{1993}}\right)$