Polynomial Division and Rational Expressions
Simplifying Complex Rational Expressions
A complex rational expression is a rational expression in which the numerator and/or the denominator are rational expressions. In other words, a complex rational expression is a nasty-looking rational expression. They are almost always written in a small font, too. Hello, eye strain.
Here are some complex—and repulsive—rational expressions. Remember that sums and differences of rational expressions are still rational expressions, and that polynomials are also rational expressions. It's too bad that polynomials can't be content being polynomials. We're hoping it's not long before it grows out of that phase where it feels the need to be somebody else.
Simplifying a complex rational expression means writing it so it looks less disgusting. By the time you're done with it, it shouldn't have any fractions in the numerator or in the denominator. Then we can make the font bigger and it won't be such a sprint to the finish line.
To simplify a complex rational expression that consists of a single fraction or polynomial in the numerator, and a single fraction or polynomial in the denominator, we do like we did with numerical fractions and use the division symbol to rewrite the expression. Good news for the division symbol, because he hasn't gotten nearly as much play since that fraction line first made an appearance. You can call him the comeback kid.
From here, all we're doing is finding the quotient of two rational expressions, which we already know how to do. Go brain, go.
Simplify the expression .
We want to make this look more appetizing, short of dipping it in marinara. We rewrite the expression as the division
From here, we know what to do. Factoring both expressions yields
Multiply the first fraction by the reciprocal of the second to find
and simplify a little bit for
Still too many parentheses for our taste, but we'll take them over a kajillion fraction lines any day.
When the numerator and/or denominator of a complex rational expression also involves addition and/or subtraction, there are three things we can do. We can
- rewrite the numerator and/or denominator so that each is a single fraction, or
- eliminate all the denominators with the help of clever forms of 1, or
- curl into a ball in the corner and sob uncontrollably.
Either of the first two methods should give the same answer, so you can use whichever you like better. You could also use both methods on the same problem as a way to check your work. You should only resort to using the third method in times of great duress. Or, you know, buy one of those squeezy stress ball things.
Rewriting the numerator and denominator
Since we know how to add and subtract rational expressions, we can collapse the numerator and denominator of a complex rational expression into single fractions...which is great for maximizing closet space, if nothing else.
We use addition to turn the numerator into a single fraction. While we could factor x2 – 1 = (x + 1)(x – 1), for this problem it's shorter not to do so. Remember, we are trying to make things simpler. We're not out on some crazed factoring binge.
We put the fractions over the common denominator (x)(x2 – 1):
Then we add and, if possible, simplify:
Now do the same sort of thing for the denominator.
We've rewritten the numerator and denominator each as a single fraction, which gives us a nicer way to write the original expression. Even though it's only slightly nicer-looking for the time being, we can already feel ourselves headed for that larger font.
From here, we rewrite the expression as a division problem and get our division on.
A complex rational expression may have denominators in the numerator and denominators in the denominator. Trust us, the numerator is none too pleased to see the denominator encroach on his territory. That's his house.
If we want to eliminate these pesky denominators right from the start and the Orkin man is unavailable, we can multiply the expression by clever forms of 1 until it looks pretty.
There's one of those irritating denominators in the numerator of the fraction—x—so we'll multiply the whole messy expression by a clever form of 1, namely x/x. This should successfully fumigate the denominator right out of there.
Now distribute x over every term in the numerator and every term in the denominator. If we bring enough for one term, we need to bring enough to share with the whole class. This gives us
Simplifying gives us
This looks better than the original expression, but we still need to eliminate the denominators x2 – 1 and x2 + 1. These things are everywhere. Someone must have left a door open.
First, knock off the x2 – 1.
Now we'll say farewell to the remaining denominator, x2 + 1.
We're almost done. We now have a rational expression that isn't complex anymore. We've finished going all Terminator on the denominators, so now we can simplify the denominator by pulling out the factor (x2 – 1) to find
We've gone from our original nightmarish expression down to the only-a-slightly-bad-dream expression
Nice work. Now drink some warm milk and try to get some shut-eye.