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Rational Expressions: Simplifying (page 2 of 3)

Sections: Finding the domain, Simplifying rational expressions

Thinking back to when you were dealing with whole-number fractions, one of the first things you did was simplify them: You "cancelled off" factors which were in common between the numerator and denominator, because dividing a number by itself gives you just "1", and you can ignore factors of "1".

So, using the same reasoning and methods, let's simplify some rationals.

• Simplify the following expression:

 

To simplify a fraction, you cancel off any common factors. Considering the factors in this fraction, I get:

Then the simplified form is: 

 

• Simplify the following rational expression: 

 

The common factor here is "x + 3", so I'll cancel that off and get:

Then the simplified form is: 

 

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 terms added 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 while looking at you sadly 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. So if the above "cancellation" is illegitimate, then so also is this one:

...and it's 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.

Always remember: You can only cancel factors, not terms!

Note: When I went from the original expression:

...to the simplified form: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

...I removed a "division by zero" problem. That is, in the original fraction, I could not have x = –3, because this would have caused division by zero. But in the reduced fraction, x was allowed to be –3. If the two expressions have different domains, can they really be equal? Not exactly. 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 the domain. That is, many (most?) books will accept the answer:

...but some will require the simplified form to have the same domain, so the 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 test).

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