- #1

- 100

- 5

## Homework Statement

Determine whether there exist ##A## and ##B## such that:

$$\frac{1}{3x^2-5x-2} = \frac{A}{3x+1} + \frac{B}{x-2}$$

## Homework Equations

None

## The Attempt at a Solution

[/B]

First I divided the polynomial ##3x^2-5x-2## by ##3x+1## and got ##x-2## as a result without a remainder, which I interpret as meaning that ##3x^2-5x-2## is the lowest common denominator of ##\frac{A}{3x+1}## and ##\frac{B}{x-2}##. Therefore what I'm looking for is:

$$\frac{A(x-2)}{(3x+1)(x-2)} + \frac{B(3x+1)}{(x-2)(3x+1)}$$

I am unsure as to how to proceed from here. Logically, it seems that we're looking for an ##A## and ##B## such that ##A(x-2) + B(3x+1) = 1##, which results in ##A = \frac{1-B(3x+1)}{x-2}##. However, I'm wondering if this is correct and/or if there's a much more obvious way to find values for A and B?

Thank you.