Polynomial Division: Simplification

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Note: Some students try to "cancel" before factorization. This cannot work! Fractions have "understood" parentheses around their numerators or denominators.

It is necessary to include these explicitly when typing fractions out sideways, such as "(2x + 4)/2", so it's clear what, exactly, is on top and what is underneath. Otherwise, the typed version would likely be mis-understood to mean "2x + 4/2 = 2x + 2", which is not what was intended.

Students may try to do this:

Even when the fractions are typeset in the math-book upright way, don't forget that there are (invisible) parentheses around the numerator and denominator, especially if the top or the bottom of the fraction has more than one term.

Instead, you must factor, and then only cancel off common factors, if any. In other words, for the exercise above, you must do the following:

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• Simplify

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Note: Most books don't talk about the domain at this point. But if your book does, you will need to note, for the above simplification, that x cannot equal zero.

Why? Because, in the original (unsimplified) form, letting x equal 0 would have caused division by zero. That's not allowed. So the original form could not allow x to equal zero. However, in the simplified form, there is no way to know about this original-form restriction. For the simplified form to be mathematically equal to the original expression, the simplified form would need to be "3x² − 5x, for all x ≠ 0".

But this is a technical point and, if your book doesn't mention anything about this now, then don't worry about it for the time being.

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• Simplify