Polynomial Division and Rational Expressions
...with the same denominator
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.
We add the numerators and keep the denominator the same, which yields
Next, we simplify by collecting like terms in the numerator to find
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 is one extra thing to be careful of: When subtracting, make sure to keep track of signs.
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 the denominator and worry about getting the numerator right. If only we didn't need to worry about the numerator, either, math would be perfect.
...with different denominators
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 to find
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 the equivalent expression where the denominator is the LCD.
First, factor the expressions to find
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
(x2 + 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.
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.