<< Chapter < Page Chapter >> Page >
This module describes the method of partial fraction expansion, in which a ratio of polynomials can be split into a sum of small polynomials. TheHeaviside cover-up method is discussed in detail with examples. Finding a partial fraction expansion in matlab is also discussed.

Splitting up a ratio of large polynomials into a sum of ratios of small polynomials can be a useful tool,especially for many problems involving Laplace-like transforms. This technique is known as partial fractionexpansion. Here's an example of one ratio being split into a sum of three simpler ratios:

8 x 2 3 x 21 x 3 7 x 6 1 x 2 3 x 3 4 x 1

There are several methods for expanding a rational function via partial fractions. These include the method of clearingfractions, the Heaviside "cover-up" method, and different combinations of these two. For many cases, the Heaviside"cover-up" method is the easiest, and is therefore the method that we will introduce here. For a more complete discussion,see Signal Processing and Linear Systems by B.P. Lathi, Berkeley-Cambridge Press, 1998, pp-24-33. Someof the material below is based upon this book.

Heaviside "cover-up" method

No repeated roots

Let's say we have a proper function G x N x D x (by proper we mean that the degree m of the numerator N x is less than the degree p of denominator D x ). In this section we assume that there are no repeatedroots of the polynomial D x .

The first step is to factor the denominator D x :

G x N x x a 1 x a 2 x a p

where a 1 a p are the roots of D x . We can then rewrite G x as a sum of partial fractions:

G x 1 x a 1 2 x a 2 p x a p

where a 1 a p are constants. Now, to complete the process, we must determine the values of these coefficients. Let's look at how to find 1 . If we multiply both sides of the equation of G(x) as a sum of partial fractions by x a 1 and then let x a 1 , all of the terms on the right-hand side will go to zeroexcept for 1 . Therefore, we'll be left over with:

1 x a 1 x a 1 G x

We can easily generalize this to a solution for any one of the unknown coefficients:

r x a r x a r G x

This method is called the "cover-up" method because multiplying both sides by x a r can be thought of as simply using one's finger to cover up this term in the denominator of G x . With a finger over the term that would be canceled bythe multiplication, you can plug in the value x a r and find the solution for r .

In this example, we'll work through the partial fraction expansion of the ratio of polynomials introduced above.Before doing a partial fraction expansion, you must make sure that the ratio you are expanding is proper. If itis not, you should do long division to turn it into the sum of a proper fraction and a polynomial. Once this isdone, the first step is to factor the denominator of the function:

8 x 2 3 x 21 x 3 7 x 6 8 x 2 3 x 21 x 2 x 3 x 1

Now, we set this factored function equal to a sum of smaller fractions, each of which has one of thefactored terms for a denominator.

8 x 2 3 x 21 x 2 x 3 x 1 1 x 2 2 x 3 3 x 1

To find the alpha terms, we just cover up the corresponding denominator terms in G x and plug in the root associated with the alpha:

1 x -2 x 2 G x x -2 8 x 2 3 x 21 x 3 x 1 1
2 x 3 x 3 G x x 3 8 x 2 3 x 21 x 2 x 1 3
3 x -1 x 3 G x x -1 8 x 2 3 x 21 x 2 x 3 4

We now have our completed partial fraction expansion:

8 x 2 3 x 21 x 2 x 3 x 1 1 x 2 3 x 3 4 x 1

Got questions? Get instant answers now!

Repeated roots

When the function G x has a repeated root in its denominator, as in

G x N x x b r x a 1 x a 2 x a j

Somewhat more special care must be taken to find the partial fraction expansion. The non-repeated terms areexpanded as before, but for the repeated root, an extra fraction is added for each instance of the repeatedroot:

G x 0 x b r 1 x b r 1 r - 1 x b 1 x a 1 2 x a 2 j x a j

All of the alpha constants can be found using the non-repeated roots method above. Finding the betacoefficients (which are due to the repeated root) has the same Heaviside feel to it, except that this time wewill add a twist by using the derivative to eliminate some unwanted terms.

Starting off directly with the cover-up method, we can find 0 . By multiplying both sides by x b r , we'll get:

x b r G x 0 1 x b r - 1 x b r 1 1 x b r x a 1 2 x b r x a 2 j x b r x a j

Now that we have "covered up" the x b r term in the denominator of G x , we plug in x b to each side; this cancels every term on the right-hand side except for 0 , leaving the formula

0 x b x b r G x

To find the other values of the beta coefficients, we can take advantage of the derivative. By taking the derivativeof the equation after cover-up (with respect to x the right-hand side becomes 1 plus terms containing an x b in the numerator. Again, plugging in x b eliminates everything on the right-hand side except for 1 , leaving us with a formula for 1 :

1 x b x x b r G x

Generalizing over this pattern, we can continue to take derivatives to find the other beta terms. The solutionfor all beta terms is

k x b 1 k x k x b r G x

To check if you've done the partial fraction expansion correctly, just add all of the partialfractions together to see if their sum equals the original ratio of polynomials.

Finding partial fractions in matlab

Matlab can be a useful tool in finding partial fraction expansions when the ratios become too unwieldy to expand byhand. It can handle symbolic variables. For example, if you type syms s , s will be treated as a symbolic variable. You can then use it as such when you make function assignments.

If you've done this and have then made a function, say H s , which is a ratio of two polynomials in the symbolic variable s , there are two ways to get the partial fraction expansion of it.A trick way is to say diff(int(H)) . When you use these functions together, Matlab gives back H expanded into partial fractions. There's also a more formal way to do it using the residue command. Type help residue in Matlab for details.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, State space systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10143/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'State space systems' conversation and receive update notifications?