Partial fractions: [2] / [ (x^2+1)^2 (x+2) ]

Question

Write following rational function in partial fraction form:

[2] / [ (x^+1)^2 (x+2) ]

Ans:

A = -2/25 B = 4/25 C = -10/25 D = 20/25 E = 2/25

Answer 1

You want to find A,B,C,D,E such that

(Ax+B)/(x^2+1) + (Cx+D)/(x^2+1)^2 + E/(x+2) = 5/((x^2+1)^2(x+2))

Placing everything over a common denominator and equating the numerators, you now get

(Ax+B)(x^2+1)(x+2) + (Cx+D)(x+2) + E(x^2+1)^2 = 5

(A+E)x^4 + (2A+B+2E)x^3 + (A+2B+C+2E)x^2 + (2A+B+2C+D+2E)x + (2B+2D+E) = 5

For the two sides of the equation to be identical, all the coefficients of each power of x must be equal. So, that means we must have

A+E = 0

2A+B = 0

A+2B+C+2E = 0

2A+B+2C+D = 0

2B+2D+E = 5

Solve those and you get A = -1/5, B = 2/5, C = -1, D = 2, E = 1/5

So the final results is (2-x)/(5(x^2+1)) + (2-x)/(x^2+1)^2 + 1/(5(x+2))