Question

Express into partial fractions 3x³ - 5x² - 3x - 40/( x² + 4)(x - 3)​

Answer 1

First simplify the rational expression by dividing. The degree in the numerator has to be at least 1 less than the degree in the denominator before you can decompose into partial fractions.

(3x³ - 5x² - 3x - 40) / ((x² + 4) (x - 3)) = 3 + (4x² - 15x - 4) / ((x² + 4) (x - 3))

Now decompose the remainder term into partial fractions:

(4x² - 15x - 4) / ((x² + 4) (x - 3)) = (ax + b) / (x² + 4) + c / (x - 3)

Multiply both sides by the denominator on the left:

4x² - 15x - 4 = (ax + b) (x - 3) + c (x² + 4)

Expand the right side:

4x² - 15x - 4 = ax² + (b - 3a) x - 3b + cx² + 4c

4x² - 15x - 4 = (a + c) x² + (b - 3a) x - 3b + 4c

Then

a + c = 4

b - 3a = -15

-3b + 4c = -4

Solve this system to get

a = 5, b = 0, c = -1

We end up with

(4x² - 15x - 4) / ((x² + 4) (x - 3)) = 5x / (x² + 4) - 1 / (x - 3)

and so

(3x³ - 5x² - 3x - 40) / ((x² + 4) (x - 3))

= 3 + 5x / (x² + 4) - 1 / (x - 3)