Fractional polynomial addition

[marksyncm]

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

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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.

 

[andrewkirk]

it seems that we're looking for an $A$ and $B$ such that $A(x-2) + B(3x+1) = 1$

That's correct, but note that that equals sign means that the equation must hold for all values of x, which can only happen if the coefficient of x on the LHS is zero. Similarly the constant term (the part that isn't multiplied by x) on the LHS must be 1. Those two requirements give you two equations, which you can solve to find the two unknown parameters A and B.