Using (u+2)^3 + 5/u^4 express x^3+5/(x-2)^4 in partial fractions
To express x^3+5/(x-2)^4 in partial fractions, we first need to make a substitution to get it in the same form as (u+2)^3 + 5/u^4. Let x-2=u. Then x=(u+2).
So we have (u+2)^3 + 5/u^4, and we want to decompose this into partial fractions. Notice that this expression has two parts. The first part, (u+2)^3, doesn't need any decomposition because it is already in its simplest form. The second part, 5/u^4, can be decomposed into partial fractions.
To decompose 5/u^4, we use the fact that any rational function with a degree of the numerator less than the degree of the denominator can be written as a sum of fractions with denominators in the form of powers of u:
5/u^4 = A/u + B/u^2 + C/u^3 + D/u^4
Now, multiplying both sides by u^4 to clear the denominators, we get
5 = Au^3 + Bu^2 + Cu + D
Now, we need to solve for the values of A, B, C, and D. Since this is an identity in u, the coefficients of the powers of u in the expression on both sides must be equal. This gives us the following system of equations:
A = 0 (coefficient of u^3 on the left side is 0)
B = 0 (coefficient of u^2 on the left side is 0)
C = 0 (coefficient of u on the left side is 0)
D = 5 (coefficient of the constant term on the left side is 5)
So we have
5/u^4 = 0/u + 0/u^2 + 0/u^3 + 5/u^4
Now, substituting back x-2 for u, we get
x^3 + 5/(x-2)^4 = (x-2)^3 + 5/(x-2)^4
The expression is already in partial fractions. Note that the expression given in the problem doesn't have non-overlapping denominators that typically result from applying partial fraction decomposition, so there isn't any further decomposition possible.