# Partial Fractions with a Repeated and a Irreducible Quadratic factor

I am trying to make this into a partial fractions form but i can't seem to find a way to do it.

The question is here:

Change into a partial fractions form.

\begin{align} \frac{2s}{(s+1)^2(s^2 + 1)} \\\\ \end{align}

i change to \begin{align} \frac{A}{(s+1)}+\frac{B}{(s+1)^2}+\frac{Cs+D}{(s^2+1)} \end{align}

I expand it and group like terms tgt but i am stump at this point \begin{align} 2s = (A+C)s^3 + (B+A+2C+D)s^2 + (A+C+2D)s + (B+A+D) \end{align}

Anyone can help?

## 4 Answers

Such partial fraction decomposition is very easy using the Heaviside cover up method. As I show in that answer, the method generalizes to quadratic denominators. Let's apply it to your problem.

$$\dfrac{a}{x+1} + \dfrac{\color{#c00}b}{(x+1)^2} + \dfrac{cx+d}{x^2+1}\, =\, \dfrac{2x}{(x+1)^2(x^2+1)}\tag{E}$$

Scaling $$\rm\,(E)\,$$ by $$\,(x+1)^2,\,$$ then evaluating at its root $$\,x = -1\,$$ yields

$$\ \ \ \ \color{#c00}b\, =\, \dfrac{2x}{x^2+1}\:\!\Bigg|_{\:\!\large x\,=\,-1} =\, \color{#c00}{ -1}$$

Subtracting out this known summand leaves a fraction in partial form, so we are done:

$$\require{cancel}\!\!\! \dfrac{\color{#0a0}{2x}}{(x+1)^2(x^2+1)}\, -\, \dfrac{\!\!\!\!\!\color{#c00}{-1}}{(x+1)^2}\: =\: \dfrac{\cancel{\color{#0a0}{2x}+x^2+1}}{\cancel{(x+1)^2}(x^2+1)}\, =\, \dfrac{1}{x^2+1}\quad {\bf\small QED}$$

Remark  It's unneeded - but instructive - to use this quadratic Heaviside method for all terms:

Scaling $$\rm\,(E)\,$$ by $$\,x^2+1\,$$ then evaluating at its root $$\,x = i\,$$ yields

$$\color{#0a0}c\, i + \color{#c00}d\, =\, \dfrac{2i}{(i\!+\!1)^2} \, =\, \dfrac{2i}{2i} \,=\, \color{#c00}1\ \Rightarrow\ \color{#0a0}{c = 0},\ \color{#c00}{d = 1}$$

Scaling $$\rm\,(E)\,$$ by $$\,(x\!+\!1)^2\,$$ then evaluating at its root $$\,x = w,\,$$ so $$\,\color{#c0d}{w^2\!+\!1 = -2w}$$

$$a(w+1)+ b\, =\, \dfrac{2w}{\color{#c0d}{w^2\!+\!1}}\,=\,\dfrac{\ \ \,2w}{\color{#c0d}{-2w}} \,=\, -1\,\Rightarrow\ a=0,\ b=-1$$

Below are some further examples asked about in comments.

$$\dfrac{f}{(x+1)^2} + \dfrac{cx+d}{x^2+1}\, =\, \dfrac{2x+1}{(x+1)^2(x^2+1)}\tag{\rm \dot E}$$

Scaling $$\rm\,(\dot E)\,$$ by $$\,(x\!+\!1)^2\,$$ then evaluating at its root $$\,x = w,\,$$

$$f\, =\, \dfrac{\color{#0a0}{2w+1}}{\color{#c0d}{w^2\!+\!1}}\,=\,\dfrac{\color{#0a0}{-w^2}}{\color{#c0d}{-2w}} \,=\, \dfrac{w}2$$

by $$\,w^2\!+2w\!+\!1=0\ \Rightarrow\ \color{#0a0}{2w\!+\!1=-w^2},\,\ \color{#c0d}{w^2\!+\!1 = -2w}$$

$$\dfrac{f}{(x-2)^2} + \dfrac{c}{x-1}\, =\, \dfrac{x}{(x-2)^2(x-1)}\tag{\rm \ddot E}$$

Scaling $$\rm\,(\ddot E)\,$$ by $$\,(x\!-\!2)^2\,$$ then evaluating at its root $$\,x = w,\,$$

$$f\, =\, \dfrac{w}{\color{c0d}{w\!-\!1}}\,=\,1+\color{#c00}{\dfrac{1}{w\!-\!1}} = 4\!-\!w$$

$$\!\underbrace{{\rm since\ we've }\,\ 0=(w\!-\!2)^2\! = (w\!-\!3)(w\!-\!1) + 1}_{\textstyle \text{polynomial}\ \color{darkorange}{\rm divide}\,\ (w\!-\!2)^2 \:\! \ {\rm by}\ \ (w\!-\!1)\qquad\ \ \ }\,\Rightarrow\, \color{#c00}{\dfrac{1}{w\!-\!1}= 3\!-\!w}$$

OR $$\bmod w\!-\!1\!:\ (w\!-\!2)^2 \equiv 1^2\,$$ so $$\,w\!-\!1\mid (w\!-\!2)^2\!\color{#c00}{-\!1^2} = \color{#c00}{(w\!-\!3)(w\!-\!1)}$$

Generally given coprime polynomials $$\,p,q\,$$ over a field $$K$$ (e.g. $$\Bbb Q, \Bbb R, \Bbb C),\,$$ we can use the Euclidean algorithm to compute partial fraction decompositions as follows

\begin{align} \dfrac{f}p + \dfrac{g}q \,&=\, \dfrac{h}{pq}\\[.4em] \iff\ q\:\!f + p\:\!g\, &=\, h\end{align}\qquad

We can solve the latter equation for $$\,f,g\,$$ using the same methods that we do for integers - by the (extended) Euclidean algorithm and closely related methods, e.g. evaluating it mod $$p\,$$ we deduce $$\,qf\equiv h\iff f\equiv h/q\,\pmod{\!p},\,$$ which is exactly the same as the Heaviside method above. $$\$$ For linear $$\,q\,$$ we can compute $$\,1/q\bmod p\,$$ simply by (long hand) polynomial $$\color{darkorange}{\text{dividing}}$$ $$\,p\,$$ by $$\,q\,$$ (as we did in the final example above), which amounts to optimizing the case when the extended Euclidean algorithm terminates in a single step.

Similar ideas were employed by Hermite in general partial fraction decomposition algorithms used for integrating rational functions, e.g. see here.

• How do you get $B$ and $C$ in $\frac{x}{(x-1)(x-2)^2} = \frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{(x-2)^2}$ using the "quadratic Heaviside method" you mentioned? Here you have $\frac{w}{w-1}$ where the denominator is linear, not quadratic. May 4, 2022 at 14:34
• Also in the original question if the numerator of the RHS is $2x+1$ instead of $2x$ then your approach to get $a$ and $b$ doesn't seem to work. May 4, 2022 at 14:37
• @NeatMath Please be more precise about what "doesn't seem to work" for you? Where are you stuck? May 4, 2022 at 14:50
• It's not clear to me how you'd arrive at $\frac{2w+1}{w^2+1}=\frac 12 (w+1) - \frac 12$ given $w^2+1=-2w$ for $\frac{2x+1}{(x+1)^2(x^2+1)}$; it's also not clear to me in the first example I mentioned above ($\frac {x}{(x-1)(x-2)^2}$): given $w^2-4w+4=0$ how do you get $\frac{w}{w-1}=-(w-2)+2$? May 6, 2022 at 15:47
• @NeatMath I appended text explaining your examples. May 12, 2022 at 10:08

Since you arrived to \begin{align} 2s = (A+C)s^3 + (B+A+2C+D)s^2 + (A+C+2D)s + (B+A+D) \end{align} identification of the coefficients of the different powers of $s$ gives $$A+C=0$$ $$B+A+2C+D=0$$ $$A+C+2D=2$$ $$B+A+D=0$$ for which the solutions are easily obtained either using matrix or elimination and the final result is : $A=0$,$B=-1$,$C=0$ and $D=1$

• You are very welcome ! May 21, 2014 at 9:43
• @user152399 It's much easier to use Heaviside - see my answer. May 21, 2014 at 15:09
• @BillDubuque. You are right but, by the end, Jyrki Lahtonen's solution is the fastest. Cheers. May 22, 2014 at 3:46
• @Claude I meant easiest in general (ad-hoc methods don't generally apply). May 22, 2014 at 4:39

Hint: It is probably easiest to observe that $$2s=(s+1)^2-(s^2+1).$$ Use that in the numerator and see what you can cancel from the two terms you get.

• Sorry, I just love spotting ad hoc tricks. May 21, 2014 at 9:31
• We can discover this (vs. pull it out of a hat) if we use the Heaviside cover-up method - see the 3rd displayed equation in my answer - where the numerators are $\,\color{#0a0}{2x} + \color{#c00}{x^2+1} = (x+1)^2\,$ when over the common denominator $(x+1)^2(x^2+1).\,$ That way it's no longer ad-hoc. May 12, 2022 at 15:20

HINT:

Using Partial Fraction Decomposition formula, $$\frac{2s}{(s+1)^2(s^2+1)}=\frac A{s+1}+\frac B{(s+1)^2}+\frac{Cs+D}{s^2+1}$$

• I tried it but i only am able to find B with Cover up rule.
– Alex
May 21, 2014 at 8:38
• @user152399, If you have four unknowns with four linear equations, right? May 21, 2014 at 9:18
• I don't understand what you mean.
– Alex
May 21, 2014 at 9:23
• @lab Using quadratic Heaviside is usually simpler than solving such systems of equations, see the Remark in my answer. May 21, 2014 at 15:44