Question

Resolve the following rational expressions into partial fractions

`(2x^2 + 5x - 11)/(x^2 + 2x - 3)`

Answer 1

`(2x^2 + 5x - 11)/(x^2 + 2x - 3)`

Since the degree of the numerator is equal to the degree of the denominator divide the numerator by the denominator

`(2x^2 + 5x - 11)/(x^2 + 2x - 3) = 2 + (x - 5)/(x^2 + 2x - 3)`   ......(1)

Consider `(x - 5)/(x^2 + 2x - 3)`

`(x - 5)/(x^2 + 2x - 3) = (x - 5)/(x^2 + 3x - x - 3)`

= `(x - 5)/(x(x + 3) - 1(x + 3))`

`(x - 5)/(x^2 + 2x - 3) = (x - 5)/((x - 1) (x + 3))`

`(x - 5)/(x^2 + 2x - 3) = "A"/(x - 1) + "B"/(x + 3)` ......(2)

`(x - 5)/(x^2 + 2x - 3) = ("A"(x + 3) + "B"(x - 1))/((x - 1) (x + 3))`

x – 5 = A(x + 3) + B(x – 1)  ......(3)

 Put x = 1 in equation (3)

1 – 5 = A(1 + 3) + B(1 – 1)

– 4 = 4A + 0

⇒ A = – 1

Put x = – 3 in equation (3)

– 3 – 5 = A(– 3 + 3) + B(– 3 – 1)

– 8 = 0 – 4B

⇒ B = 2

Substituting the values of A and B in equation (2) we have

`(x - 5)/(x^2 + 2x - 3) = (-1)/(x - 1) + 2/(x + 3)`

∴ The required partial fraction is

`(2x^2 + 5x - 11)/(x^2 + 2x - 3) = 2 + 2/(x + 3) - 1/(x - 1)`