Polynomials and Partial Fractions Decomposition

Question

In a bid to check my understanding I have this case: Decompose: $ \dfrac{x^4-8}{x^2+2x}$ Here I see it is an improper factions- degree of $x$ is higher on the numerator than on denominator. Using long division to divide, I get $$x^2-2x+4+ \dfrac{(-8x-8)}{x^2+2x}$$

this is also same as $$x^2-2x+4+ \dfrac{(-8x-8)}{x(x+2)}$$ How do I proceed to derived the decomposition template fraction?

Answer 1

Let the last fraction term be $\dfrac{A}{x}+\dfrac{B}{x+2}$, then

$$\begin{align} \frac{x^4-8}{x^2+2x} =& x^2 - 2x + 4 + \frac{-8x-8}{x^2+2x}\\ =& x^2 - 2x + 4 + \frac{A}{x}+\frac{B}{x+2}\\ =& x^2 - 2x + 4 + \frac{A(x+2)+Bx}{x(x+2)}\\ \end{align}$$

By comparing coefficients on the numerator of the fraction term, $$A=-4,\ B=-4$$

So $$\frac{x^4-8}{x^2+2x} = x^2 - 2x + 4 - \frac{4}{x} - \frac{4}{x+2}$$

Answer 2

for numerical working there is a useful rule which applies when the denominator polynomial is a product of distinct linear factors. suppose $f(x)=\frac{P(x)}{Q(x)}$ where $deg(Q)=n \gt deg(P)$ and $$Q(x) = \prod_{k=1}^n (x-\alpha_k) $$ where the $\alpha_k$ are all different. then define $Q_k(x) = \frac{Q(x)}{(x-\alpha_k)}$

in these happy circumstances we may write: $$ \frac{P(x)}{Q(x)} = \sum_{k=1}^n \frac{P(\alpha_k)}{Q_k(\alpha_k)} (x-\alpha_k)^{-1} $$

in the example given we have $\alpha_1=0$ and $\alpha_2=-2$ giving $Q_1(x)=x+2$, $Q_1(\alpha_1) = 2$ and $Q_2(x)=x$, $Q_2(\alpha_2) = -2$ with $P(\alpha_1)=P(0)=-8$ and $P(\alpha_2)=P(-2)=8$. you will see that this gives the answer already obtained.

don't be put off by the explication - with a little practice this is a very straightforward procedure where it is applicable. for example this can be written straight down:

$$ \frac{x^2+x+1}{(x-1)(x-2)(x-3)} = \frac{\frac32}{(x-1)} - \frac7{(x-2)} + \frac{\frac{13}2}{(x-3)} $$