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Now that we know what rational expressions are, we can investigate general polynomial division. Any polynomial division problem can be written as a rational expression, and any rational expression can be interpreted as a polynomial division problem. For example, could be read as the rational expression "8x + 9 over 3x2," or it could be read as the polynomial division "8x + 9 divided by 3x2." That fraction line is pulling double-duty. We hope it's compensated accordingly.

To do polynomial division when we don't have the guarantee that things will work out evenly, we use the same techniques we used earlier for dividing polynomials. We also cross our fingers and hope that it will work out evenly anyway.

  1. Cancel any common factors that show appear in both the numerator and denominator.
  2. Factor and cancel any factors that appear in both the numerator and denominator.
  3. Use long division.

There's also a neat trick we can sometimes use if we want to feel clever. Yes, even more clever than usual.

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