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Polynomial Division and Rational Expressions
Rational Expressions
Rational expressions are nothing to be afraid of. Don't let the bared fangs and red, glowing eyes fool you. Despite the scary drooling, they're only a particular type of expression. The eyes are a hereditary condition.
We can do anything with rational expressions that we can do with other expressions. We can evaluate them, simplify them, add, subtract, multiply, and divide them. We can eat them in a box. We can eat them with a fox. We can even eat them off of day-old gym socks—wait, that wasn't how that rhyme went? Sorry, Doc.
When we do arithmetic with rational expressions, we have rational expressions for answers. A sum, difference, product, or quotient of two rational expressions is a rational expression. Quoth the Shmaven, "as long as we don't try to divide by 0, bro."
Rational expressions are incredibly similar to rational numbers (fractions). For one, they both have one of those line thingies in the middle. It makes sense that arithmetic with rational expressions is similar to arithmetic with fractions. If you're comfortable with fraction arithmetic, this analogy will be helpful. If you aren't comfortable with fraction arithmetic, the best way to use your time right now would be to get better acquainted with fractions and come back here later. Spoon if you need to. We won't tell anyone.
Sometimes the quotient of two integers is another integer. For example,
It's nice when this happens, because we don't need to worry about remainders or decimals or anything like that. Let's be honest, nobody likes a remainder. We might make polite small talk with them at dinner parties, but as soon as their back is turned...
However, the quotient of two integers is often not an integer. That's not good. In order to make sense out of something like
,
we need rational numbers.
Similarly, sometimes the quotient of two polynomials is a polynomial. Here's an example:
.
Again, it's nice when this happens, and you know how to find this sort of quotient by pulling out a common factor, factoring a polynomial, or using long division. You love doing things you know how to do, because it means maybe we haven't been wasting your time. However, the quotient of two polynomials often isn't a polynomial. Ugh. We really didn't want to say that.
In order to make sense out of something like
,
we need rational expressions. If we're going to need these things, we'd better find out what they are.
A rational expression is an expression of the form
.
Basically, a rational expression is like a rational number (a fraction), except instead of having one integer over another we have one polynomial over another. So "rational" means one thing over another. In other words, we can turn it into a "ratio." Ahhh. That makes sense.
We still call the polynomial on top the numerator and the polynomial on the bottom the denominator, mainly because mathematicians were tired of naming things and had run out of cool sounding names. We also still interpret the fraction bar as division, so the denominator still isn't allowed to be 0, or equivalent to 0. Sorry, denominator. You abused your 0 privileges, and now you've had them taken away.
Sample Problems
- is a rational expression, since this is one polynomial over another.
- x^{-1} is a rational expression, since and both 1 and x are polynomials.
- is a rational expression, since 3 and 4 are both (constant) polynomials.
- x^{2} + 3x + 4 is a rational expression, since we could write this as and 1 is a nonzero polynomial.
The moral of example 4 above is that any polynomial is also a rational expression, in the same way that any integer is also a rational number. We can stick any polynomial in the numerator of a fraction and put 1 in the denominator. This may make 1 feel a little worthless. Well, maybe now he knows how 0 feels most of the time.
Non-Examples
- is not a rational expression, since 2^{x} is not a polynomial...as much as it wishes it were.
- is not a rational expression, since 2 – 2 = 0 and we aren't allowed to have 0 in the denominator.