00:01
So we have a function for s minus 16 over s plus 2 times s squared plus 4.
00:10
And we want to find the partial fraction expansion for this.
00:14
So this is going to be a over s plus 2 plus bs plus c over s squared plus 4.
00:24
And when you have the quadratic, irreducible quadratic factor, you have to do the linear factor.
00:30
So we're going to essentially multiply by the denominator s plus 2 times s squared plus 4.
00:39
So this will distribute to each term killing the denominator here so we get 4s minus 16.
00:47
The s plus 2 square will cancel out with this so we get a times s squared plus 4 plus e s plus 3 whoops, c, times the s squared will cancel, s squared plus 4 will cancel out with this term here, so it's going to be bs plus c, or bs plus c times s plus 2.
01:14
Let's distribute, s squared plus 4a plus bs squared, right, distribute, then it's going to be plus 2 bs, s is this term, now we get c, s, and then lastly 2c.
01:38
All right, so now let's pair this, let's find our like terms.
01:41
So let's look at the s squared.
01:43
So there's s squared here, s squared here, so we have zero equals a s squared plus b s squared.
01:52
Now let's look at the s terms.
01:55
We got four s, we've got two b s and c s.
01:59
So we've got 4s equals 2es plus cs.
02:06
And then last, we'll do in this purple color, let's look at the constants.
02:12
We have negative 16, 4a, and 2c.
02:17
16 equals 4a plus 2c.
02:20
And you'll know you've done this correct if you have the same number of terms on the right side as you do on all the right sides.
02:25
So one, two, three, four, five, six.
02:27
One, two, three, four, five, six.
02:30
There we go.
02:31
So let's look at this first one here.
02:36
Well if we're looking at this we can cancel out the s's and just deal with a and b.
02:44
So we have 0 equals a plus b which means that a is equal to negative b.
02:51
Now let's look at this equation here.
02:58
Well this is already b and c.
03:00
Or b and c, excuse me.
03:02
But we can get rid of the s's, let's just get rid of them.
03:06
So it's four equals two b plus c.
03:10
Now for this term here, this has a and c, but we wanna do it with b and c...