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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 remainders. We might make polite small talk with them at dinner parties, but as soon as their backs are 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.
It's the same deal with polynomials. 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.
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.
Evaluating a rational expression is exactly like evaluating any other expression. We substitute values for variables, do some arithmetic, and see what we find. It's usually much prettier than what we were previously dealing with, because we don't have all those ugly x's and y's hanging around by the end of it. However, with rational expressions, there's one thing to be careful of: we aren't allowed to put in values that make the denominator 0, because then we'd be dividing by 0 and the world would come crashing down.
Chicken Little thought things were bad when that acorn beaned him in the noggin. Watch out, Chicken Little. You'd better keep your eyes peeled for a whole other breed of ovoid.
Evaluate the rational expression at each given value of x.
1. x = -3
All we've gotta do is plug in x = -3 and do some math.
Ah, beautiful.
2. x = 0
Second verse, same as the first. Replace all those x's with 0s to get:
Remember, it's fine if one of our variables equals 0, as long as the denominator doesn't. If we have nothing on the bottom, we'll have some real problems. We also won't be able to visit any public beaches.
3. x = 2
Psych! We aren't allowed to do this, because we'll have (2)^{2} – 4 = 0 in the denominator. See? We told you: no dividing by 0 allowed. When, oh when, will we learn?
When we're given a rational expression, we can simplify it by doing the first two steps of polynomial division: we cancel any common factors, then factor the numerator and denominator and cancel anything else we can get away with. You can try to factor and cancel other things that you probably won't get away with, too, but the penalties are severe if you happen to be caught. That's x – x + 10 years in prison.
There is one weird thing we need to remember, though. The good news is that it's weird, which should make it easy to remember.
Be careful: When you simplify a rational expression, you may find a different expression than the one you started with.
This needs some clarification. Thankfully, that's what we do.
Simplifying a fraction gives us an equivalent fraction; and refer to the same quantity, so they're the same number, right? Just because the second fella is puffing out his chest doesn't mean his value is any greater. If that's the case, why would simplifying a rational expression give us a different expression?
The reason is that we're not allowed to evaluate a rational expression for values that make its denominator zero. Whoa. Deja vu.
When we simplify a rational expression, the values we're allowed to plug in may be different in the simplified expression than they were in the original expression.
If we factor the expression , we find .
If we cancel x + 1 from the numerator and denominator, we get x – 1.
If we evaluate the expressions and x – 1 at any value of x except x = -1 we'll find the same answer. However, we can't evaluate at x = -1. This would make the denominator 0, and that would cause the heads of mathematicians all over the world to start spinning violently around in circles. Instead, to demonstrate that we got this, we write:
This shows that we know how to simplify rational expressions and that we understand the weirdness of simplifying rational expressions. The first step to understanding the weirdness of simplifying rational expressions is acceptance. Followed by bargaining.
Remember that two expressions are called equivalent if they evaluate to the same number for every possible value of the variable(s). The example above shows two expressions that look like they should be equivalent, but aren't, in the same way that not every pair of similar-looking human beings are twins.
A rational expression and its simplified version may or may not be equivalent. We must be vigilant.
As you'll recall from that time you became the grandmaster lord high poobah of all things fractions, multiplying fractions requires us to multiply the numerator by the numerator and the denominator by the denominator.
Simple stuff. After multiplying numerators and denominators, we simplify if there's any simplifying to be done. Even simpler.
Sometimes we can simplify each fraction before we multiply them.
To multiply , first simplify the individual fractions to find .
That will save us some serious pencil graphite in the long run. Think of how many dozens of pennies you could save over the course of your lifetime if you follow this rule. Seriously though, it'll make your life easier, even if it doesn't make you a pencil baron. When the fractions involve big numbers, simplifying can be annoying, and you don't need that kind of aggravation. How irked would you be if you had to deal with a baddie like
Thankfully, we don't need to bother finding 7 × 121. We don't even need to pay someone to find it for us. Instead, we factor the numbers first and multiply second:
Now we can see which factors cancel, plain as day:
This was almost fun. We did say "almost," though, so let's not get too crazy here. We were able to cross out a lot of stuff and get a nice answer without having to bother with multiplication. That's at least as entertaining as a game of Boggle.
We can do similar things with rational expressions. To multiply two rational expressions, we multiply the numerators and multiply the denominators. We'll see if we can have even more fun this time around. Ladies and gentlemen, don your party hats.
Multiply
While we could multiply out the numerator and the denominator for quadratic polynomials, why bother? We've already spent enough time on our quads today. Besides, that's another step of arithmetic where mistakes can sneak in, assuming you're not perfect. If your teacher is okay with answers in this form, don't make extra work for yourself. Do the bare minimum, and continue to scrape by in life until someone notices.
Just kidding! We're kidding so hard.
Be careful: When asked to multiply two rational expressions, factor them first. This will let you quickly and easily see which factors cancel out. It'll also give you an opportunity to wear your "Factor Fiction?" T-shirt.
Dividing rational expressions is exactly like multiplying them, except that we turn the second expression upside-down. Make sure the lid is sealed tightly before you attempt this. You don't want your variables going everywhere.
Remember how dividing fractions works? If we want to divide two fractions, we multiply the first fraction by the reciprocal, or multiplicative inverse, of the second fraction.
To find we multiply by the reciprocal of :
We're sure it comes as a big surprise that this is how division with rational expressions works as well. We can even picture you making your "surprised" face.
As with multiplication, we factor expressions before we divide them so that we can see which factors cancel out. Let's try it out.
First we factor:
Then we change division to multiplication by taking the reciprocal:
Finally, we cancel factors wherever possible:
One other tip that's useful for division of rational expressions is that any polynomial can be written as a rational expression by putting the polynomial over 1. Note: polynomials under 1 are probably still on soft food or formula. That information is neither here nor there, but we thought that you should know.
Adding or subtracting rational expressions with the same denominator is like adding or subtracting fractions with the same denominator. We add or subtract the numerators and keep the denominator the same. This arrangement is fine with the denominator, who likes himself just the way he is. Good for him.
Addition of rational expressions is relatively straightforward. We add the numerators and simplify by collecting like terms. If you ever have a friend over, though, don't ask if they want to look at your term collection. Personal experience tells us that they probably won't be interested.
Find .
We add the numerators and keep the denominator the same.
Next, we simplify by collecting like terms in the numerator.
That's it. So easy a caveman could do it. Great, now we'll be hearing from those guys in the Geico commercials.
With subtraction, though, there's one extra thing to be careful of: make sure to keep track of any rogue minus signs.
Find .
Subtracting the numerators gives us:
Notice that we're subtracting the whole chunk (x – 4), so we need to be careful with signs when we simplify. We don't want to blow this. Hm? Oh...blow chunks. Very funny.
The nice thing about adding or subtracting rational expressions with the same denominator is that we don't need to think about the denominator. We copy it and worry about getting the numerator right. If only we didn't need to worry about the numerator, either, math would be perfect.
When asked to add or subtract rational expressions with different denominators, we first need to find the least common denominator (LCD) of the expressions. After turning each expression into an equivalent expression where the denominator is the LCD, we can add or subtract as we did earlier. We like being able to do things we did earlier, because it means we need to learn less new stuff. Our brains are getting full.
To find the LCD of a pair of fractions, we first factor the denominators. The LCD must contain every factor from each denominator. The number of times a factor appears in the LCD must be the same as the largest number of times the factor appears in any one denominator. Don't try shorting the LCD. It will know, and it won't be happy about it. Plus, it knows a guy.
First, we'll do this with number-type fractions.
Find the LCD of and . Rewrite the fractions as equivalent fractions where the denominators are the LCD.
To find the LCD, first we factor the denominators of the two fractions.
The factors in the denominators are 5 and 6, so the LCD must have factors of 5 and 6. Since the factor 5 occurs twice in one of the denominators, the factor 5 must also occur twice in the LCD. The LCD of the two fractions is:
5 × 5 × 6 = 150
To rewrite a fraction so that it has the LCD as a denominator, we multiply the fraction by a clever form of 1. The clever form of 1 uses the factors of the LCD that aren't in the denominator yet. Clever, right? Like a fox.
To do this with rational expressions instead of rational numbers, we need to factor polynomials instead of numbers. It's a good idea to simplify each rational expression first to keep the LCD as simple as possible. We don't want the LCD to be complicated, because then we'll have that Avril Lavigne song stuck in our head all day. Again.
Find the LCD of the rational expressions .
For each expression, write an equivalent expression where the denominator is the LCD.
First, factor the expressions:
We can simplify the first expression, so the rational expressions we'll deal with going forward are:
The LCD must contain every factor in either denominator. Since none of these factors occurs more than once, we don't need to do any fancy-shmancy finagling and the LCD is simply all three factors multiplied together:
(x^{2} + 2)(x + 3)(x + 2)
There's no reason to do extra work, so we'll leave the LCD like that. Someone else can come along and clean it up if they'd like. To write a rational expression over a common denominator, we multiply the expression by a clever form of 1. The clever form of 1 needs to use the factors that are in the LCD but not in the fraction's denominator yet. Gosh, so many rules. What is this, Soviet Russia?
Now that we know how to find LCDs, we can add and subtract rational expressions that have different denominators. First, we put the rational expressions over the same denominator. Then, we add or subtract them according to what the problem tells us to do. If the problem tells us to jump off a cliff, we do that too, but only off a very short cliff. Gotta be smart about these things.
Add .
First we find the LCD of the two expressions, which is (x + 1)(x + 2). The equivalent rational expressions are:
Now that we have expressions with the same denominator, we can do the addition:
This expression simplifies to , which is our final answer. Unless your teacher asks you to multiply out the denominator, don't bother. If your teacher asks you to jump off a cliff, tell her the problem beat her to it, and you're already on it.
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