More Worked Examples & A Trick with Subtractions

Faced with a polynomial fraction like this, many students will try to do something like the following:

Is this legitimate? Can the student really do this? (Think "bleeding, oozing...") You can NOT cancel term-by-term! You can ONLY cancel factors!

So the first thing I have to do — if I'm to have any chance of doing this simplification correctly — is factor the numerator and the denominator:

Since the numerator and denominator share a common factor, I can reduce the expression as:

Can I reduce any further? For instance, can I cancel off the x's? (whimpering, bleeding...) Can I cancel a 2 out of the 4 and the 6? (oozing, flopping...) No! This is as simplified as it's going to get, because there are no remaining common factors. Then the answer is:

The factors in the numerator and denominator are almost the same, but not quite, so they can't be cancelled — yet. If the fraction had been:

...(that is, if the signs between the terms had been plusses instead of minuses), I could have rearranged the terms as:

...and cancelled to get 1, since order doesn't matter in addition. But order most definitely does matter in subtraction, so I can't just flip the subtraction to get matching factors. However, take a look at this:

Do you see what happened? When I reversed the subtraction in the second line, I got the same number but the opposite sign. Then I can flip a subtraction, as long as I remember to change the sign; I'll flip the subtraction and kick out a "minus" sign.

So it turns out that I can reverse one of the subtractions in the original rational expression above, as long as I remember to switch the sign out front:

Now I can cancel:

Where did the 1s come from? Remember: If "nothing" is left, then a 1 is left. So, putting it all together:

I can't cancel the x's; I can't cancel the 3 into the 12. I cannot reach inside (the parentheses that are understood to be around) the numerator and denominator, rip some pieces out, and try to cancel. (...big brown eyes...)

No; to simplify this, I first need to factor. Only once I have factors can I see if I there are any common factors that can be cancelled off.

Then my answer, accounting for the division-by-zero issue that I removed, is:

To simplify this, I need to factor. Also, I can see that I will need to flip the subtraction in the denominator; I'll need to remember to change the sign when I do that flip.

Then my hand-in answer is: