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I am trying to figure out this simple problem but I have no idea how to do it.

$\frac {2x^3 - 5x^2 -4x -3}{2x^3 + x^2 - 18x -9}$

I know that I can keep guessing at random to find a factor in the terms to factor them out but that seems like it would be far too time intensive, especially considering that if this was on a test it would likely be the first step of the problem and I can't spend 30 minutes just setting the problem up.

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You can try the polynomial long division algorithm which is very reminiscient of regular long division. check for an example – MSEoris Dec 2 '12 at 22:49
up vote 2 down vote accepted

Simply in what sense? Removing common factors?

There are only 8 things to try to find a rational root of the numerator. It shouldn't take 30 minutes to work out what those 8 are and to test them. (note that it can be proven that if they have a common factor, then not only does the factor have integer coefficients, but both cubics also have a rational root)

Or, you can use the Euclidean algorithm to find the (polynomial) greatest common divisor of your two polynomials.

e.g. to find the gcd of $x^2 - 2x + 1$ and $x^2 - 1$, you can compute:

$$ \begin{align} (x^2-2x+1, x^2-1) &= (x^2 - 2x + 1 - (x^2 - 1), x^2 - 1) \\ &= (-2x + 2, x^2 - 1) \\ &= (x-1, x^2 - 1) \\ &= (x-1, x^2 - 1 - x(x-1)) \\ &= (x-1, x-1) \\ &= x-1 \end{align}$$

(note the "size" of a polynomial here is determined by its degree, and I was "lazy" and avoided doing a full division-with-remainder calculation)

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