How do you write the partial fraction decomposition of the rational expression #(3x^2 + 4x) / (x^2 +1)^2#?

Question

Serenity Jones · Accepted Answer

The answer is #=(3)/(x^2+1)+(4x-3)/(x^2+1)^2#

Let's do the decomposition in partial fractions

#(3x^2+4x)/(x^2+1)^2=(Ax+B)/(x^2+1)+(Cx+D)/(x^2+1)^2#

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

Therefore,

#(3x^2+4x)=((Ax+B)(x^2+1)+(Cx+D))#

Let #x=0#, #=>#, #0=B+D#

Coefficients of #x^2#

#3=B#

Coefficients of #x#

#4=A+C#

Coefficients of #x^3#

#0=A#

#C=4#

#=B+D#

#D=-3#

So,

#(3x^2+4x)/(x^2+1)^2=(3)/(x^2+1)+(4x-3)/(x^2+1)^2#

Greyson Brown · Answer

undefined To write the partial fraction decomposition of the rational expression (3x^2 + 4x) / (x^2 + 1)^2, follow these steps:

1. Factor the denominator: (x^2 + 1)^2 can be factored as (x^2 + 1)(x^2 + 1).

2. Since the denominator has repeated linear factors, the partial fraction decomposition will contain linear and quadratic terms.

3. Write the expression in the form of partial fractions, where each term in the decomposition has a numerator that is a polynomial of degree one less than the corresponding factor in the denominator:

(3x^2 + 4x) / (x^2 + 1)^2 = A/(x^2 + 1) + B/(x^2 + 1)^2

4. Find the values of A and B by equating the original expression to the partial fraction decomposition and solving for A and B.

5. Multiply through by the denominator to clear the fractions and solve for A and B:

3x^2 + 4x = A(x^2 + 1) + B

6. Expand the right side and group like terms:

3x^2 + 4x = Ax^2 + A + B

7. Equate coefficients of like terms:

For x^2 terms: 3 = A
   For x terms: 4 = 0 + A

8. Solve for A:

A = 3

9. Substitute the value of A back into the equation and solve for B:

3x^2 + 4x = 3(x^2 + 1) + B
   3x^2 + 4x = 3x^2 + 3 + B

10. Equate coefficients:

4 = 3 + B

11. Solve for B:

B = 1

So, the partial fraction decomposition of the given rational expression is:

(3x^2 + 4x) / (x^2 + 1)^2 = 3/(x^2 + 1) + 1/(x^2 + 1)^2