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Multiplying Rational Expressions (page 1 of 2)

Remember how you multiply regular fractions: You multiply across the top and bottom, and cancel off any duplicate factors. For instance:

You always need to simplify, where possible:

While the above is perfectly valid, it is generally simplest to cancel first and then multiply, since you'll be dealing with smaller numbers that way:

This process (cancel first, then multiply) works with rationals, too.

• Simplify the following expression:

Simplify by cancelling off duplicate factors:

Then the answer is:

Why did I add "for x not equal to 0"? Because the domain of the original function did not include x = 0 (since this would have caused division by zero). For the two expressions to be technically equal, their domains have to be the same. Since ^3x/₂ has no problem at x = 0, I have to explicitly state this exclusion. Your text might not make this distinction. If you're not sure if your teacher cares about this technicality, make sure you find out before the test.

• Multiply and simplify the following:

Many students find it helpful to convert the "15" into a fraction. This can make it a little more obvious what cancels with what.

Can you cancel off the 2 into the 20? No! When you have a fraction like this, there are understood parentheses around any sums of terms, like this:

You can only cancel off factors (the entire contents of a set of parentheses), not terms (one of the addends inside a set of parentheses). Going inside the parentheses and hacking x's and y's and arms and legs off the poor polynomial doesn't simplify anything; it just leaves the little polynomial lying there on the floor, quivering and bleeding and oozing and whimpering...

Okay, maybe not; but you get the point: Never reach inside the parentheses and hack off part of the contents. Either you cancel off the entire contents with a matching factor from the other side of the fraction line, or you don't cancel anything at all. (I told you I'd be making a big deal of this!)

The only thing that factors out of the 20x + 25 is a 5, and that doesn't cancel off with the 2 underneath, so, for this rational, there is no further reduction to be done. Then the final answer is:

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• Multiply and simplify the following expression:

Some students, when faced with this problem, will do something like this:

Can you really "cancel" like this? (Think "bleeding"...) Is this even vaguely legitimate? (Oozing...) Has this student done anything at all correctly? (Flopping, whimpering...) No, no, and no! You can not cancel terms; you can only cancel factors.

So my first step has to be to factor. Once I've factored everything, I can cancel off any factor that is mirrored on either side of the fraction line. The legitimate simplification looks like this:

Can I now cancel off the 2's? (Tears are welling up in the polynomial's eyes...) Can I cancel off the x's with the x²? (Now the polynomial is starting to cry...) Hmm??? No! The x's are only part of their factors; they are not stand-alone factors, so they can't cancel off with anything.

Then the answer, including the trouble-spots that I removed from the domain when I cancelled the common factors, is:

The "x not equal to 0, –1 or –3" came from the factors that I cancelled off; your book may not require this information.

Note: For reasons which will become clear when you are adding rational expressions, it is customary to leave the denominator factored, as shown above. At this stage, your book may or may not want the numerator factored. You should recognize, in any case, that "(2x – 5)(x + 2)²" is the same thing as "2x³ + 3x² – 12x – 20", and convert the form of your answer if your book or instructor expects it.

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