Chapter 8, Problem 14PS

Partial Fraction Decomposition Suppose the denominator of a rational function can be factored into distinct linear factors

$D(x)=(x-c_1)(x-c_2)\cdots (x-c_n)$

for a positive integer n and distinct real numbers $c_1,c_2,\dots ,c_n$.If .V is a polynomial of degree less than n. show that

$\frac{N(x)}{D(x)}=\frac{P_1}{x-c_1}+\frac{P_2}{x-c_2}+\cdots +\frac{P_n}{x-c_n}$

where $P_k=N(c_k)/D^k(c_k)$ for $\begin{array}{l}\\ k=1,2,\dots ,n\end{array}$, Note that this

is the partial fraction decomposition of $N(x)/D(x)$.