Find the partial fraction decomposition.

Question

Find the partial fraction decomposition.

3x^2 +11x/ x^2 +6x+9

Answer 1

Long division:

3

__________________

x^2 + 6x + 9 | 3x^2 + 11x

3x^2 + 18x + 27

_____________________

-7x - 27

THe denominator factors (x+3)^2

-7x-27 A B

----------- = ------------ + ------------

(x+3)^2 (x+3) (x+3)^2

-7x-27 = A(x+3) + B

-7x-27 = Ax +3A + B

-7 = A

-27 = 3A+B

-27 = 3(-7) + B

-27 = -21 + B

B = -6

PFD:

3 -7 -6

+ ---------- + ----------

(x+3) (x+3)^2

Check:

the numerator is 3(x+3)^2 + -7(x+3) + -6

= 3(x+3)^2 + -7x - 21 + -6

= 3(x+3)^2 +-7x + -27

= 3(x^2+6x+9) + -7x-27

= 3x^2 + 18x + 27 - 7x -27

= 3x^2 + 11x

YES, it checks