Question

Find the sum of the fourth powers of the roots of
$x^3-2x^2+x-1=0$.

Answer 1

Here $f\left(x\right)=x^3-2x^2+x-1$,
$f^{\prime }\left(x\right)=3x^2-4x+1$.
Also $\frac{f^{\prime }\left(x\right)}{f\left(x\right)}=\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}$
$=\sum (\frac{1}{x}+\frac{a}{x^2}+\frac{a^2}{x^3}+\frac{a^3}{x^4}+....)$
$=\frac{3}{x}+\frac{S_1}{x^2}+\frac{S_2}{x^3}+\frac{S_2}{x^4}+....$;
Hence $S_4$ is equal to the coefficient of $\frac{1}{x^5}$ in the quotient of $f^{\prime }\left(x\right)$ by $f\left(x\right)$ which is very conveniently obtained by the method of synthetic division as shown:
Hence the quotient is $\frac{3}{x}+\frac{2}{x^2}+\frac{2}{x^3}+\frac{5}{x^4}+\frac{10}{x^5}+....$;
Thus $S_4=10$.