Simplifying Rational Expressions

Using the same reasoning and methods, let's simplify some rational expressions.

How do you simplify rational expressions?

To simplify rational expressions, first completely factor the polynomials in the numerator and the denominator. If there are any factors that are common to the numerator and denominator (that is, if you've got stuff on top and underneath that match), cancel off these factors.

Depending on the class and the context, you might be expected to take whatever is left and multiply it back together. At this stage, though, leaving things factored is probably fine. When you get to adding rational expressions, you'll probably multiply out the numerators, but leave the denominators factored. This is becuase, once you have a common denominator, you'll be adding the numerators, so it will be helpful to have added terms, rather than multiplied factors, when doing that addition.

• Simplify the following expression:

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• Simplify the following rational expression:

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Warning: The common temptation at this point is to try to continue on by cancelling off the 2 with the 4. But you cannot do this. Whenever you have an expression containing terms that are added(or subtracted) together, there are understood parentheses around them, like this:

You can only cancel off factors (that is, entire expressions contained within parentheses), not terms (that is, not just part of the contents of a pair of parentheses).

To go inside the parentheses and try to cancel off part of the contents is like ripping off arms and legs of the poor little polynomial trapped inside. It'll be bleeding and oozing and flopping around on the floor, whimpering plaintively while sadly gazing up at you with big brown eyes...

Well, okay; maybe not. But trying to cancel off only a portion of a factor would be like trying to do this:

Is 66/63 equal to 2? Of course not. And if the above "cancellation" is illegitimate, then so also is this one:

...and this is illegitimate for exactly the same reason as the previous one was. While it isn't quite so obvious that you're doing something wrong in the second case with the variables, these two "cancellations" are not allowed because you're reaching inside the factors (the 66 and 63 above, and the x + 4 and x + 2 here) and ripping off *parts* of them, rather than cancelling off an entire factor.

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There is one technical consideration which is often overlooked in algebra, but crops up later in calculus. In the exercise above, when I went from the original expression:

...to the simplified form:

...I removed a "division by zero" problem. That is, in the original fraction, I could not have plugged in the value x = −3, because this would have caused division by zero.

But in the reduced fraction, x was allowed to be −3. So the question is, if the two expressions have different domains, can they really be equal?

No, they're not exactly equal. To be exactly equal, they must have the same domains (and ranges).

Depended upon the text you're using, this technicality with the domain may be ignored or glossed over, or else you may be required to make note of it. Specifically, many (most?) textbooks will accept the following as your answer:

...but some books (and instructors) will require that your simplified form be adjusted, as necessary, in order to have the same domain as the original form, so the technically-complete answer would be:

Depending on your book and instructor, you may not need the "as long as x isn't equal to −3" part. If you're not sure which answer your instructor is expecting, ask now, before the next test.