...so after all that you now have the technology (all right, the skills) to factorise a polynomial completely. You first do trial and error to find the first linear factor, then you multiply it by ax2 + bx + c and compare coefficients to find the quadratic. Then you check if the quadratic factorises further.
Onto dividing polynomials proper....
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It is based on the same mathematics (the distributive law), but it is useful for polynomials with more than two terms. Not only that, but knowing this easy method for multiplying leads to fewer mistakes, and it is also the basis for a great method for dividing one polynomial by another.
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