00:01
In this question, we are asked to find the series expansion for the given function using partial fractions first.
00:08
And by the partial fraction decomposition, we want this to be equal to a over x minus 5 plus b over x plus 4, where a and b are some numbers.
00:24
And to find them, we need to bring the expression on the right hand side to the common denominator.
00:30
And to do that, we need to multiply the first fraction by x plus 4.
00:34
And the second fraction we need to multiply it by x minus 5.
00:42
And we will get a times x plus 4 plus b times x minus 5 divided by x minus 5 times x plus 4.
01:03
Now let's expand the expression in the numerator and collect the term, combine the terms with x together.
01:12
We are going to get a x plus bx and collect the terms without x together too.
01:18
4a minus 5b and finally this becomes a plus b times x plus 4a minus b minus 5b now compare this to the to the expression at the very beginning they have same denominator and for two expressions to be equal they must have same numerators and to have same numerators the coefficients in front of different powers of x and the three coefficients must be equal.
02:16
The coefficient in front of x on the left -hand side is 1, and the coefficient in front of x on the right -hand side is a plus b.
02:28
So i want 1 to be equal to a plus b.
02:34
The three coefficient on the left -hand side is 58, and the three coefficient on the right -hand side is 4a -5b.
02:42
So i want 58 to be equal to 4 -a -5b.
02:44
Minus 5b this gives us a system of equations for a and b from the first equation a equals to 1 minus b and now we are going to plug in this a in the second equation to get 58 equals to 4 times 1 minus b minus 5b this becomes 58 equals to 4 minus 4 b minus 5b is going to be minus 9b.
03:38
And then this simplifies to negative 9b equals to what to 54.
03:54
And then after dividing by negative 9, we are going to get that b equals to b equals to negative 6.
04:02
And then a equals to 1 minus negative 6...