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I'm sure you are all familiar with partial fraction decomposition, but I seem to be having trouble understanding the way it works. If we have a fraction f(x)/[g(x)h(x)], it seems only logical that it can be "split up" into A/g(x) + B/h(x) for some A and B, because there must be some A and B that make this work, right? But I don't see why this wouldn't work when we have f(x)/[g(x)]^2. Couldn't we just split it like A/g(x) + B/g(x) for some A and B? I am missing something, but I don't know what. Could someone please enlighten me as to why my reasoning is incorrect? Thank you.

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You need parentheses in the first, as you have shown $\frac{f(x)h(x)}{g(x)}$ Looks like your first example should be $\frac{A}{g(x)}+\frac{B}{h(x)}$, and the second should be $\frac{A}{g(x)}+\frac{B}{g(x)}$ –  Ross Millikan Jun 27 '11 at 20:11
    
Did you mean to ask why we can't split it as $A/g(x)+B/g(x)$? That could be simplified to the single fraction $(A+B)/g(x)$, which would equal the original expression only if $f(x)$ were equal to $(A+B)g(x)$. –  Brian M. Scott Jun 27 '11 at 20:12
    
"split it like A/f(x) + B/f(x)" should be "like A/g(x) + B/g(x)", I think. –  Arturo Magidin Jun 27 '11 at 20:12
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Do you know any linear algebra? There's a very nice explanation if you know about dimensions of vector spaces and the size of their bases. –  MartianInvader Jun 27 '11 at 22:40
    
@Arturo Magidin, Ross Millikan, and Brian M. Scott: All of you are correct and I will edit my post above accordingly. Thank you. –  Hautdesert Jun 28 '11 at 0:28
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6 Answers 6

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Well, of course you could split it up as $$ \frac{f(x)}{(g(x))^2} = \frac{A}{g(x)}+\frac{B}{g(x)} $$ (with non-constant $A$ and $B$) but what you find wouldn't be very useful, since any A and B that satisfy $$ A + B = \frac{f(x)}{g(x)} $$ are a solution. And if it were easy to solve that, you would divide by $g(x)$ first without trying to get partial fractions (also the solutions to that are never unique).

Instead, by splitting it up as $$ \frac{f(x)}{(g(x))^2} = \frac{A}{g(x)}+\frac{B}{(g(x))^2} $$ You get your solutions to: $$ A (g(x))^2 + B g(x) = f(x) $$ Which will usually give you unique terms for polynomial $f,g$.

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Another way of looking at Arturo's comments about unique factorization: the key to the standard partial-fraction decomposition is the existence of the Euclidean Algorithm for polynomials, along with the standard-but-unspoken assumption that $g(x)$ and $h(x)$ in your initial decomposition have no factor in common - that is, that their GCD is 1. If we can find a partial-fraction decomposition for $1\over g(x)h(x)$ as ${A(x)\over g(x)}+{B(x)\over h(x)}$, then we can certainly find one for $f(x)\over g(x)h(x)$ for any polynomial $f$; just multiply by $f(x)$ in that decomposition. Likewise, if we can find a partial-fraction decomposition for $f(x)\over g(x)h(x)$ for any polynomial $f$, then we can certainly find one for the special case $f(x)\equiv 1$. But saying that ${1\over g(x) h(x)} = {A(x)\over g(x)}+{B(x)\over h(x)}$ is the same as saying that ${A(x)h(x) + B(x)g(x) \over g(x)h(x)} = {1\over g(x)h(x)}$, or in other words saying that $A(x)h(x) + B(x)g(x) \equiv 1$, and the existence of polynomials $A$ and $B$ with these properties is exactly what the (extended) Euclidean algorithm provides. It doesn't work for your case of $f(x)\over g^2(x)$ because $GCD(g(x),g(x)) \neq 1$, so there can't be any $A(x)$ and $B(x)$ satisfying $A(x)g(x)+B(x)g(x) \equiv 1$.

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Note that the above is simply an elaboration of the equivalences in my answer. –  Bill Dubuque Jun 27 '11 at 21:18
    
@Bill: Yeah, your answer appeared while I was working on mine; I do think (hope!) the added details help, though... –  Steven Stadnicki Jun 27 '11 at 21:43
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No, in general you cannot split it that way. For example, $$\frac{x}{(x+1)^2}$$ cannot be written as $$\frac{A}{x+1} + \frac{B}{x+1}$$ with $A$ and $B$ constants. Because you would just get $$\frac{A}{x+1} + \frac{B}{x+1} = \frac{A+B}{x+1},$$ with $A+B$ constant, so this is definitely not equal to $\frac{x}{(x+1)^2}$.

If you have $$\frac{A}{g(x)} + \frac{B}{g(x)}$$ then the answer is just $$\frac{A+B}{g(x)},$$ just like with fractions, you have $$\frac{A}{n} + \frac{B}{n} = \frac{A+B}{n}.$$ So unless $\frac{f(x)}{(g(x))^2}$ can be simplified by cancelling one of the factors of $g(x)$, you have no hope of writing it as a sum of two fractions with $g(x)$ in the denominator.

(Somewhat hidden in the above is the fact that polynomials also have unique factorization, so if $$\frac{f(x)}{g(x)} = \frac{h(x)}{k(x)}$$ then $f(x)k(x) = g(x)h(x)$. If $f(x)$ and $g(x)$ have no common factors and $h(x)$ and $k(x)$ have no common factors, then the equality means that you must have $g(x)=k(x)$ and $f(x) = h(x)$ up to multiplication by constants; again, just like an equality of fractions $$\frac{a}{b} = \frac{c}{d}$$ with $a,b,c,d$ all integers, and $a$ and $b$ have no common factors and $c$ and $d$ have no common factors, then up to sign you must have $a=c$ and $b=d$.)

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Unique factorization plays no role above. $\rm\:f/g^2 = h/g \iff f = g\:h\:$ is true in any ring where $\rm\:g\:$ is a unit. –  Bill Dubuque Jun 27 '11 at 22:04
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HINT $\rm\displaystyle\qquad \frac{1}{g\:h}\ =\ \frac{a}g\ +\ \frac{b}h\ \iff\ 1\ =\ a\ h\: +\: b\ g\ \iff\ 1\: =\: \gcd(g,h)$

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There is a simple closed formula for the partial fraction decomposition:

Let $X$ be an indeterminate. For any complex number $a$, any nonnegative integer $k$, and any rational fraction $f\in\mathbb C(X)$ defined at $a$, let $$T_a^k(f)$$ be the degree at most $k$ Taylor approximation of $f$ at $a$.

Let $N,D\in\mathbb C[X]$ be polynomials. Assume $$ D(X)=\big(X-a_1\big)^{m_1}\cdots\big(X-a_r\big)^{m_r}, $$ where the $a_j$ are distinct and the $m_j$ positive. Let $f$ be the rational fraction $N/D$. Then we have $$ f(X)=Q(X)+\sum_{j=1}^r\ T_{a_j}^{m_j-1}\Big(f(X)\big(X-a_j\big)^{m_j}\Big)\big(X-a_j\big)^{-m_j} $$ for a unique polynomial $Q$. If $m_j=1$ for all $j$, we get Lagrange's Interpolation Formula $$ f(X)=Q(X)+\sum_{j=1}^r\ \frac{N(a_j)}{X-a_j}\ \prod_{k\not=j}\ \frac{1}{a_j-a_k}\quad. $$

If $\deg N < \deg D$, then $Q=0$. Otherwise, putting $q:=\deg N-\deg D$, we have $$ Q(X^{-1})=T_0^q\Big(f(X^{-1})X^q\Big)X^{-q}. $$ The above expressions for $f$ are called partial fraction decomposition.

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If you have $\frac {f(x)}{g(x)^2h(x)}$ the standard partial fraction decomposition would be:

$\frac {a(x)}{g(x)} + \frac{b(x)}{g(x)^2} + \frac {c(x)}{h(x)} + d(x)$

The assumption would be that $g(x)$ and $h(x)$ are coprime, and most applications I've seen have them irreducible. Then the degrees of $a(x)$ and $b(x)$ are less than the degree of $g(x)$ and the degree of $c(x)$ is less than the degree of $h(x)$.

Of course the fractions involving $g(x)$ can be put over a common denominator of $g(x)^2$, but the point of the standard decomposition is that the degrees of the numerators are less than the degree of $g(x)$ and not that of $g(x)^2$.

Similar observations apply if there is more than one squared term in the denominator, or if higher powers are involved.

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