While doing some self-study from the book: Higher Algebra by Hall & Knight, I encountered some articles that I could not understand properly. Those articles were Art. 562 & Art. 563.
Art 562: The result of the preceding article(given below) enables us very easily to find the sum of an assigned power of the roots of an equation.
Result of preceding article: $f'(x)=\frac{f(x)}{(x-a)}+\frac{f(x)}{(x-b)}+....+\frac{f(x)}{(x-k)}$,
where a,b,....,k are the roots of the equation $f(x)=0$
Art 563: When the coefficients are numerical we may also proceed as in the following example.
Here is where the problem lies
Example: Find the sum of the fourth powers of the roots of
$$x^3-2x^2+x-1=0$$
Here $f(x)=x^3-2x^2+x-1$, $f'(x)=3x^2-4x+1$. $$\frac{f'(x)}{f(x)}=\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}$$ $$=\sum{\Bigl(\frac{1}{x}}+\frac{a}{x^2}+\frac{a^2}{x^3}+....\Bigl)\qquad\qquad...(1)$$ $$=\frac{3}{x}+\frac{S_1}{x^2}+\frac{S_2}{x^3}+\frac{S_3}{x^4}+.....;\qquad\qquad...(2)$$
hence $S_4$ is equal to the coefficient of $\frac{1}{x^5}$ in the quotient of $f'(x)$ by $f(x)$, which is very conveniently obtained by the method of synthetic division as follows:
Hence the quotient is $\frac{3}{x}+\frac{2}{x^2}+\frac{2}{x^3}+\frac{5}{x^4}+\frac{10}{x^5}+.....;$ thus $S_4=10$.
I could not understand the method of synthetic division and the steps (1) and (2).
Any help will be appreciated.
For all $x$ such that $|x|>\max (|a|,|b|,|c|)$ and for each $d\in \{a,b,c\}$ we have $$\frac {1}{x-d}=(1/x)\cdot\frac {1}{1-d/x}=(1/x)\sum_{n=0}^{\infty}(d/x)^n=$$ $$=(1/x)+(d/x^2)+(d^2/x^3)+...$$ If we sum this over each of the 3 series with $d=a, d=b$ and $d=c,$ the resulting co-efficient of $x^{-5}$ is $a^4+b^4+c^4.$
Imagine that $f(x$) and $f'(x)$ are representations of numbers in number-base $x,$ although we allow the digits (the co-efficients) to be any numbers, not just integers from $0$ to $x-1.$ Synthetic division follows the same rules of long division that you would use for computing all of the digits (co-efficients) of all integer-powers of of the base, including especially the negative powers.