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.
Sample Problem
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.
Example 1
Simplify. Is the simplified expression equivalent to the original, or is it frontin'?
|
.
.
at 
but we cannot evaluate the original expression at 
. Don't take our word for it, try it and see. The verdict is in: frontin'. We knew it.Example 2
Simplify. Is the simplified expression equivalent to the original?
|
.
.Exercise 1
Simplify the following and determine if the simplified expression is equivalent to the original: 
Exercise 2
Simplify the following and determine if the simplified expression is equivalent to the original: 
.Exercise 3
Simplify the following and determine if the simplified expression is equivalent to the original: 
.
, but the simplified expression can be evaluated at 
. This means the two expressions are not equivalent. Bummer. We had such high hopes for these two.Exercise 4
Simplify the following and determine if the simplified expression is equivalent to the original: 
Exercise 5
Simplify the following and determine if the simplified expression is equivalent to the original: 
, which simplifies to x + 1. While x + 1 can be evaluated for any value of x, the original expression can't be evaluated at x = 0 or 
, so the two expressions aren't equivalent.