Partial Fraction Decomposition (cont'd)

Question

Find the P.F.D. of the following function.

Answer 1

27/(x-1)^2(x+2)^3

= a/(x-1) + b/(x-1)^2 + c/(x+2) + d/(x+2)^2 + e/(x+2)^3

= 1/(x-1) + 1/(x-1)^2 +1/(x+2) + 2/(x+2)^2 + 3/(x+2)^3 = 27

multiply both sides by (x-1)^2(x+2)^3

= (x-1)(x+2)^3 + (x+2)^3 + (x-1)^2(x+2)^2 + 2(x-1)^2(x+2) + 3(x-1)^2 = 27(x-1)^2(x+2)^3

expand, combine like terms

27x^5+106x^4+12x^3-252x^2-96x+213 = 0

graph, find the x intercept = about -2.94093= the only real solution. all other 4 solutions are imaginary

Answer 2

I'm new to Wyzant and wasn't sure how to attach a file here. I've provided a step-by-step solution, which you can access using the following link:

https://github.com/lasik23/wyxant/blob/main/Partial%20Fraction%20Decomposition.pdf

Answer 3

27/[(x-1)²(x+2)³] = A/(x-1) + B/(x-1)² + C/(x+2) + D/(x+2)² + E/(x+2)³

27 = A(x-1)(x+2)³ + B(x+2)³ + C(x-1)²(x+2)² + D(x-1)²(x+2) + E(x-1)²

If x = 1, then 27B = 27. So, B = 1.

If x = -2, then 9E = 27. So, E = 3.

So, 27 = A(x-1)(x+2)³ + (x+2)³ + C(x-1)²(x+2)² + D(x-1)²(x+2) + 3(x-1)²

If x = 0, then -8A + 8 + 4C + 2D + 3 = 27. So, -8A + 4C + 2D = 16.

if x = -1, then -2A + 1 + 4C + 4D + 12 = 27. So, -2A + 4C + 4D = 14

if x = 2, then 64A + 64 + 16C + 4D + 3 = 27. So, 64A + 16C + 4D = -40

Solve the system: -8A + 4C + 2D = 16

-2A + 4C + 4D = 14

64A + 16C + 4D = -40

Subtract the second equation from the first: -6A - 2D = 2

Add -4 times the first equation to the third equation 96A -4D = -104

So, 108 A = -108. A = -1

6 - 2D = 2. So, D = 2

Since -2A + 4C + 4D = 14, A - 2C - 2D = -7

So, -1 -2C -4 = -7

-2C = -2 So, C = 1

A = -1, B = 1, C = 1, D = 2, E = 3