SOLUTION: How would I solve this polynomial division problem? (24x^3 + 3) / (2x + 1)

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Question 292165: How would I solve this polynomial division problem?
(24x^3 + 3) / (2x + 1)
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Well, you have a couple of choices:
1) You could divide using long division or synthetic division.
2) You could try factoring the numerator and see if anything will cancel the denominator.
Let's try 2)
%2824x%5E3%2B3%29%2F%282x%2B1%29 Factor 3 from the numerator.
3%288x%5E3%2B1%29%2F%282x%2B1%29 Now factor the binomial in the numerator as the sum of two cubes.
3%288x%5E3%2B1%29+=+3%28highlight_green%28%282x%29%5E3%2B%281%29%5E3%29%29
Recall that the sum of two cubes will factor thus:
A%5E3%2BB%5E3+=+%28A%2BB%29%28A%5E2-AB%2BB%5E2%29 and here, A = 2x, B = 1, so...
3%288x%5E3%2B1%29+=+3%282x%2B1%29%284x%5E2-2x%2B1%29 so now we have:
3cross%28%282x%2B1%29%29%284x%5E2-2x%2B1%29%2Fcross%28%282x%2B1%29%29=3%284x%5E2-2x%2B1%29+=+highlight%2812x%5E2-6x%2B3%29