00:01
Okay, so we're asked to find a part of the fraction.
00:05
Let's look at the denominator.
00:10
Well, yes, we can.
00:11
That's just x squared minus 4 times x squared minus 4.
00:17
And we can factor this more.
00:19
We get x minus 2 times x plus 2 times x minus 2 times x plus 2.
00:29
Okay, so that's squared times x plus 2 squared.
00:35
So we have two linear factors that are repeating.
00:40
So these are both case 2s.
00:42
Okay, so let's write that out.
00:45
We have 3x squared plus 12x minus 20, all over x minus 2 squared, then x plus 2.
01:00
Okay, so that's to stay over x minus 2, plus b over x minus 2 to the power of 2.
01:09
Plus c over x plus 2 c over x plus 2 the power 2 okay if we multiply by the common denominator which is x minus 2 squared times x plus 2 squared we get 3x squared x minus 20 is equal to a times x minus 2 x plus 2 to 5 2 plus b times x plus 2 to prior 2 plus c times x plus 2 times x minus 2 squared and lastly we have plus d which is x minus 2 squared and expanding the right hand side we get a times x cubed plus 2 x squared minus 4 x plus right now that's minus eight plus b times x squared minus or that's plus four x plus four plus c times x cubed and it's two x squared minus four x plus eight and lastly we have plus d times x squared minus four x plus four x plus four okay now expanding this more, we get a x cubed, plus 2x squared, minus 4.
03:16
I forgot the 2a, x squared, minus 4ax, minus 4a, plus b, x squared, plus b, plus b, plus c, cubed, minus 2 ,8, no, c, x squared, minus 4 ,000, let's see, c, x squared, minus 4 ,000, x plus x plus hc plus d x squared minus 4 d x plus 4d now combining life terms let's start with x cubed x cubed we have a plus c x cubed and then our x squared terms here here here and here and here and here and here and here and that's 2 a plus 2c plus d, x squared.
04:26
Now our x terms right here, here, here, and here.
04:33
So we have negative 4a plus 4b, minus 4c, minus 4t, x, and then our constant terms, which is 4d, negative a, a, 4b, and 8c, that's negative 8a, plus plus 4d plus 8c plus 4d.
05:03
Okay, so this is our left hand side and this is our right hand side.
05:09
So let's rewrite this as a system.
05:11
Let's see, so we have a coefficient in front of x cubed, which is a plus b.
05:17
And then on our left hand side, so it's a plus c is equal to so.
05:23
And now the coefficient in front of x squared, x3 is equal to 2a plus plus b minus 2c plus d, 2a plus b, minus 2c plus b.
05:45
And now the coefficient in front of x, that's 12, is equal to negative 4a plus 4b, minus 4c, minus 4d.
06:05
And then our constant term is negative 8a plus 4b, plus 8c plus 4d is equal to negative 20.
06:22
So looking at these four equations, i can see that i can simplify a bit.
06:29
I'm going to divide this term by four and this term by four.
06:35
What i get? so i get negative a plus b minus c, minus d is equal to three.
06:46
And we get here a negative two, a plus b plus d is equal to negative 5.
07:01
Okay, let's see what we can do.
07:03
Can we? so i want another equation in terms of a and c.
07:10
Right now i have equation one in terms of a and c...