Partial Fraction Decomposion - Free Educational videos for Students in K-12

Partial Fraction Decomposion - By The Organic Chemistry Tutor

Transcript


00:00	in this video , we're going to go over partial
00:02	fraction decomposition . I'm going to show you how to
00:05	solve many of these problems . But before we do
00:07	that , let's talk about what partial fraction decomposition is
00:13	. So , what exactly is it ? Well ,
00:15	to illustrate it , let me give you an example
00:17	problem , let's say if we have to over three
00:20	X plus 4/5 wire , how can we combine these
00:26	two fractions Into one single fraction while we need to
00:31	get common denominators first . So I'm going to multiply
00:34	the second fraction by three X over three X .
00:37	And the first one By five Y over five wide
00:41	. So this will give me 10 y Over 15
00:45	x . Y plus 12 X Over 15 XY .
00:52	Now , because I have a common denominator , I
00:56	can combine it into a single fraction , so I
00:58	can write it as 10 Y plus 12 X ,
01:02	Divided by 15 x . y . Now , with
01:06	partial fraction decomposition allows me to do Is it allows
01:10	me to take a single fraction like this one And
01:13	break it down into two smaller fractions . So it's
01:18	the reverse of combining two fractions into a single fraction
01:22	. So you take a single fraction and break it
01:25	down into multiple smaller fractions . It could be two
01:28	fractions 34 It depends on the nature of this particular
01:33	fraction . And so that's the basic idea behind partial
01:36	fraction decomposition . So let's work out an example ,
01:40	let's say we have a rational function seven x -23
01:46	divided by X squared minus seven X plus 10 .
01:52	So how can we take this fraction and break it
01:55	down into smaller fractions . The first thing we need
01:59	to do is we need to factor this fraction completely
02:04	. We can't factor the numerator . However , we
02:06	can factor this binomial X squared minus seven X plus
02:10	10 . So what ? two numbers multiply to 10
02:13	But add to the middle coefficient -7 . So we
02:17	know five times two is 10 but negative five plus
02:19	negative two adds up to negative seven . So we
02:23	have seven x monastery divided by x minus two And
02:29	X -5 . So you need to write the denominator
02:33	in terms of linear factors and quadratic factors . So
02:37	what exactly is a linear factor ? Linear factors are
02:41	like X plus two , three X -5 . four
02:46	x plus eight X . These are linear factors .
02:50	A quadratic factor . Well look something like this ,
02:55	X square plus eight X plus tree which is not
03:00	fact herbal or it could be X squared plus seven
03:06	. These are quadratic factors . Now there are some
03:11	other terms that you need to be familiar with .
03:14	Mhm . How would you describe this term ? This
03:22	is known as a repeated linear factor . So X
03:26	squared is also a repeated linear factor . Seven X
03:30	plus three squared . That's a repeated linear factor now
03:34	X squared plus one square . That's a repeated quadratic
03:38	factor . Be familiar with these terms because it's going
03:42	to affect the way this fraction is decomposed . So
03:46	we have to linear factors and what we need to
03:50	write is two fractions on this side and each fraction
03:56	will contain a linear factor . Now , anytime you
04:00	have a linear factor on the bottom you need to
04:02	put a constant on top . Now we're going to
04:05	choose two different constants A and B . Our goal
04:08	is to determine A and B . It's a equal
04:11	to three is be equal to four . We need
04:14	to figure that out . So what I'm gonna do
04:17	is I'm going to multiply both sides of the equation
04:21	by this denominator That is by X -2 And X
04:27	-5 . So if we multiply this fraction by what
04:31	we have here X -2 and X -5 or cancel
04:36	giving us seven X -23 on the left side .
04:43	Now if I take this fraction and multiply it by
04:46	what I have here , The X -2 terms will
04:49	cancel . Leaving behind a times X -5 . Now
05:00	these two will cancel and I'm gonna be left with
05:02	B times X -2 . Now when you get to
05:06	this part there's two ways in which you could find
05:09	and be . You could use a system of linear
05:13	equations or you can plug in X . Values and
05:17	determine A . And B . For a simple problem
05:21	like this , it's best to plug in X values
05:24	. So let's try plugging in X equals five because
05:28	if we do so This will disappear . 5 -5
05:31	is zero and 0 times 80 . So then this
05:35	becomes seven X -23 or seven times 5 -23 .
05:40	Because we need to replace that with five , that's
05:44	equal to a Times 5 -5 plus B . Times
05:49	5 -2 . seven times 5 is 35 and 5
05:56	-50 . So this disappears and 5 - to a
06:00	street Now 35 -23 is 12 . And so that's
06:05	equal to three B . And if we divide both
06:07	sides by three B is equal to four . So
06:10	I'm just going to rewrite that here . So now
06:13	we need to calculate a so this time let's plug
06:25	into so if Acts is equal to two let's see
06:29	what's going to happen . So we're gonna have seven
06:31	times 2 -23 And that's equal to a times 2
06:37	-5 Plus B . Times 2 -2 . seven times
06:43	2 is 14 and 2 -5 it's negative three And
06:49	2 -2 is zero . Now 14 -23 That's a
06:55	negative nine And that's equal to -38 B times zero
06:59	is 0 so that disappears . And now let's divide
07:02	both sides by -3 . So -9 divided by -3
07:06	history . And so that's the value for A .
07:13	Now let's go back to our original problem . So
07:23	we had seven x -23 divided by x minus two
07:30	Times X -5 . And that was equal to a
07:35	over X -2 Plus B Over X -5 . Now
07:42	we have the value of A . A . History
07:45	and be it's four . So therefore we could see
07:51	that this fraction Is equal to three over X -2
07:57	plus four Over X -5 . That's the answer .
08:02	Now , if you ever need to check it ,
08:05	simply combine the two fractions and see if they give
08:08	you what you started with here . So let's go
08:11	ahead and try that . I'm going to multiply this
08:13	fraction by X -2 Over X -2 . Whatever you
08:17	do to the top you have to do to the
08:18	bottom and this one I'm going to multiply by X
08:21	-5 , Divided by X -5 , Three times x
08:27	-5 is three X -15 . And on the bottom
08:31	we have X -5 Times X -2 which I'm going
08:35	to leave it that way And then four times X
08:38	-2 , That's gonna be four x minus eight Divided
08:43	by X -5 X -2 . Now If we add
08:48	three x and four x That will give us seven
08:52	x . And then if we add negative 15 a
08:55	negative eight That will give us a -23 . So
09:00	as you can see the answer that we had at
09:04	the beginning was correct . So it's three over X
09:08	-2 . Plus four over X -5 . That's the
09:12	answer . Now . For the sake of practice ,
09:15	let's try another similar problem . So let's say maybe
09:20	about that . If we want to decompose 29 monastery
09:24	acts divided by X squared -1 -6 , feel free
09:34	to pause the video . Go ahead and try that
09:38	. So first we need to factor denominator by the
09:41	way , notice that the denominator is always one degree
09:44	higher and then the new mayor , now , what
09:47	? Two numbers multiply is too negative six . And
09:50	that's the middle coefficient , -1 . This is going
09:53	to be -3 . And positive too negative three plus
09:56	two adds up to negative one . So we have
09:59	29 -3 acts divided by Ex Monastery Times X-plus two
10:06	. So let's set this equal to a over X
10:09	. Monastery plus B Over X-plus two . Since we
10:15	have two linear factors . Now , just like before
10:23	we're going to multiply both sides of this equation by
10:26	this phenomena . So that is by X -3 times
10:30	x plus two . So these two will cancel .
10:34	And on the left side We're going to get 29
10:38	minus three X . Now if we take a over
10:46	X minus tree and multiplied by this , we can
10:50	see that X minus three will counsel and then we're
10:56	going to get A times X plus two . And
11:05	then if we take this fraction multiplied by X .
11:08	Monastery times X plus two , The X-us two factors
11:12	will cancel . Leaving behind be times acts Monastery .
11:21	Actually , let's keep that there for now . So
11:25	at this point let's plug in X equal stream .
11:30	This is going to be 29 monastery time street and
11:33	that's equal to A . Times three plus two plus
11:38	B . Times stream . Monastery Dream ministry is nine
11:44	and three plus 2 is five , 3 -3 is
11:48	zero . So this disappears 29 -9 is 20 And
11:52	that's equal to 5 8 . So if we divide
11:56	both sides by five , we can see that A
12:02	. Is equal to four . I'm going to put
12:12	that in the side over here . Yeah . Now
12:23	let's plug in negative too . So that this term
12:25	becomes zero . So we're gonna have 29 -3 times
12:32	negative two and that's equal to a -2 plus two
12:37	Plus B . Times -2 . Monastery . Now negative
12:43	three times negative two is positive six , negative two
12:48	plus two is zero and negative two minus three is
12:50	negative five , 29 Plus six is 35 . And
12:55	so that's equal to -5 . B . So B
12:58	is going to be 35 divided by negative five Which
13:02	is -7 . So now that we have A .
13:09	And B , we cannot write the final answer .
13:12	So let's plug it in to this expression . So
13:15	29 over I mean 29 minus three . Acts over
13:19	X squared minus x minus six . That's equal to
13:22	four divided by X . Monastery . And then B
13:26	is negative seven . So instead of R . N
13:28	plus negative seven , I could simply right minus seven
13:34	because a positive number of times a negative number it's
13:37	still a negative result . So it's gonna be minus
13:39	seven over X plus two and this is the answer
00:0-1	.

Summarizer

DESCRIPTION:

This precalculus video tutorial provides a basic introduction into partial fraction decomposition. The full version of this video contains plenty of examples and practice problems with repeated linear factors and repeated quadratic factors. Partial fraction decomposition is the process of taking a complex fraction and breaking it into multiple simpler fractions. It's the reverse of adding combining two fractions into a single fraction.

OVERVIEW:

Partial Fraction Decomposion is a free educational video by The Organic Chemistry Tutor.

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