Question

Ezpress 3x + 11/x² + 6x + 9
as partial fractions,​

Answer 1

`(3)/(x + 3) + (2)/((x+3)^2)`

Explanation:

Given

`(3x + 11)/(x^2 +6x + 9)`

Required

Express as partial fraction

`(3x + 11)/(x^2 +6x + 9)`

Expand the numerator

`(3x + 11)/(x^2 +3x +3x+ 9)`

Factorize

`(3x + 11)/(x(x +3) +3(x+ 3))`

Factor out x + 3

`(3x + 11)/((x +3)(x+ 3))`

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

As a partial fraction, we have:

`(3x + 11)/((x +3)^2) = (A)/(x + 3) + (B)/((x+3)^2)`

Take LCM

`(3x + 11)/((x +3)^2) = (A(x+3) + B)/((x + 3)^2)`

Cancel out (x + 3)^2 on both sides

`3x + 11 = A(x+3) + B`

Open bracket

`3x + 11 = Ax+3A + B`

By comparison, we have:

`Ax = 3x` ===>
`A = 3`

`3A + B = 11`

Substitute 3 for A

`3*3 + B = 11`

`9 + B = 11`

Solve for B

`B = 11-9`

`B =2`

Substitute:
`A = 3` and
`B =2` in

`(3x + 11)/((x +3)^2) = (A)/(x + 3) + (B)/((x+3)^2)`

`(3x + 11)/((x +3)^2) = (3)/(x + 3) + (2)/((x+3)^2)`

Hence, the partial fraction is:

`(3)/(x + 3) + (2)/((x+3)^2)`