Question

How do you write the partial fraction decomposition of the rational expression `(3x + 2) / [(x - 1)(x + 4)]`?

Answer 1

`(3x+2)/((x-1)(x+4)) = 1/(x-1) + 2/(x+4)`

Explanation:

Since the denominator already in the factor form we can rewrite it as

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

Multiply both sides by the LCD to get

`3x+ 2 = A (x+4) + B(x-1)`

Multiple the expression to get

`3x+ 2 = Ax + 4A + Bx -B`

Set up the system of equation like this

`x: " " " " A + B = 3`
`x^0 " " " " 4A -B = 2`

Solve system by elimination method

`A + B = 3`

`4A -B = 2`

`5 A " " = 5"`

`=>A= 1`

Solve for B

`(1) + B = 3 => B = 2`

Partial fraction of `(3x+2)/((x-1)(x+4)) = 1/(x-1) + 2/(x+4)`