00:01
In this problem, we're being asked to find the partial fraction decomposition of our given fraction.
00:05
Well, in order to do this, we have to first start by factoring our denominator.
00:10
So to do that, i'm going to take our denominator rated off to the side here.
00:14
So i'm going to do all this work on the side.
00:16
So we're going to have 2x of the 3rd minus x squared minus 8x plus 4.
00:21
And we need to factor this.
00:23
Well, there is no greatest common factor for the whole polynomial.
00:26
Now, there are 4 terms, though, so i'm going to factor by grouping.
00:30
To do this, i'm going to kind of make a barrier here.
00:32
And i'm going to look for the greatest common factor for those first two terms, which is x squared.
00:37
So i'm going to factor out on x squared, and then i'd be left with 2x minus 1.
00:42
Now, for my second two terms, i have a gcf of negative 4.
00:46
So i'm going to factor out of negative 4, and then i'm also left with 2x minus 1.
00:51
So now that both of these terms have this common factor of 2x minus 1, i can rewrite this as the quantity of 2x minus 1, and then i'm left with x squared minus four.
01:03
So times the quantity of x squared minus four.
01:06
But take a look at that second factor, x squared minus four.
01:10
Well, that's just a difference of two squares.
01:12
So when we factor this, we're going to bring down to x minus one.
01:16
And when we factor x squared minus four, we're going to get x plus two and x minus two.
01:21
And now we have factored our denominator.
01:24
So i'm going to come over here and i'm going to rewrite my fraction as 9.
01:29
X squared minus 9x plus 6 all over the quantity of 2x minus 1 times the quantity of x plus 2 times the quantity of x minus 2.
01:43
So now let's look at our factors.
01:46
Well, they are all distinct linear factors.
01:50
So because of that, for our first fraction, when we do our partial fraction decomposition, we'll simply be a over 2x minus 1.
01:59
Our second fraction will be b over x plus.
02:01
2, and our third fraction will be c over x minus 2.
02:07
And remember, this will all be equal to 9x squared minus 9x plus 6 all over the quantity of 2x minus 1 times x plus 2 times x minus 2.
02:22
So now we've rewritten our basic partial fraction decomposition.
02:27
And remember, what we now have to do is we now have to get, find the values of a, b, and c.
02:33
Well, as you can see, i've ran out of room.
02:36
So i'm going to start a new page here.
02:39
So remember, we just found that we have a over 2x minus 1 plus b over x plus 2 plus c over x minus 2.
02:51
Whoops, forgot the minus there.
02:55
And it's all equal to 9x squared minus 9x plus 6 all over the quantity of 2x minus x minus x minus 1 times x plus 2 times x minus.
03:07
So now what i'm going to do is multiply both sides of our equation by our common denominator.
03:15
So our common denominator is 2x minus 1, x plus 2, and x minus 2.
03:22
And i'm going to do the same thing on the right hand side.
03:25
We're going to multiply it by 2x minus 1, x plus 2, and x minus 2.
03:30
Now i typically wouldn't write it vertically, but as you can see, i don't really have that much room.
03:35
So now, for my first fraction, we have a over 2x minus 1.
03:39
So that means it's missing the factors x plus 2 and x minus 2.
03:43
So i need to multiply a by x plus 2 and x minus 2.
03:50
Now for my second fraction, i have b over x plus 2.
03:53
So it's missing the 2x minus 1 and x minus 2 factors.
03:58
So we're going to have b times 2x minus 1 and x minus 2.
04:03
Our third fraction is c over x minus 2.
04:07
So it's missing the 2x minus 1 and x plus 2 factors.
04:10
So i'm going to have c times 2x minus 1 times x plus 2.
04:16
And now on the right -hand side of our equation, remember the reason why we multiplied by our common denominator is so that way this fraction cancels out and we're just left with the numerator.
04:26
So it's all equal to 9x squared minus 9x plus 6.
04:32
Now there's a couple ways you can go about solving this.
04:35
Now, one thing we can do is think about the factors that we have and figure out when they're equal to 0.
04:41
Because then we can substitute those values into this equation and solve for one of the variables.
04:47
Well, our first factor, in that first term, i have x plus 2.
04:51
Well, it's 0 would be when x is equal to negative 2.
04:55
So here's what i can do.
04:57
I'm going to take when x is negative 2, and i'm going to substitute x equals negative 2 into this equation.
05:03
So i'm going to have a times, well, negative 2 plus 2 is 0, and negative 2 minus 2 is 0.
05:11
Or sorry, is negative 4.
05:14
Then i'm going to have b times, well, if i substitute negative 2 into 2x minus 1, i get negative 5.
05:21
And when i substitute it into x minus 2, i get negative 4.
05:26
Then i'm going to have plus c times.
05:29
Well, if i substitute negative 2 into 2x minus 1, i'm going to get negative 5...