00:01
Let's find the partial fraction decomposition for both of these functions, and there's no need to determine the numerical values of the coefficients.
00:11
So looking at the first fraction, the first thing that i noticed is that the numerator is a higher degree than the denominator.
00:20
So anytime the numerator has degree equal to or larger than the denominator, the first thing we should always do is long division or polynomial division.
00:30
And that's what we have set up over here on the right.
00:34
So we'd like to find the quotient here.
00:37
So we have an x squared, x to the fourth.
00:40
So we've got an x squared up here.
00:43
And then we multiply 2x cubed plus x squared.
00:50
And then we subtract the whole thing.
00:55
Doing so, we have a lot of cancellation here.
01:04
And then we're just left with 2x minus 1.
01:08
Now this quadratic that we're dividing by does not go into 2x minus 1.
01:14
Minus 1.
01:16
So we have for part a we can rewrite this as the quotient which is x squared and then we have the remainder over the thing that we originally divided by.
01:39
This is why the long division is useful.
01:43
Now every time we finish long division, the denominator will have larger degree than the numerator and now we do partial fraction to composition.
01:52
So as before, the first thing, always always look at that denominator, see if it can be factors.
01:59
Here we can factor this is x minus 1 squared.
02:05
So this is what the book calls case 2...