Simplifying and Multiplying Rational Expressions

On tuesday we started out the class everyone's favorite way - a surprise homework collection! But once everyone was done stressing about how they did, we moved onto a new topic, simplifying rational expressions. This is similar to simplifying fractions, but with variables involved. You can cancel out any terms that the numerator and denominator have in common, just like one would if you were simplifying even the simplest of fractions, such as 5/20. 5 and 20 are both divisible by five, so if you divide both by five the fraction simplifies to 1/4.

Once we added variables to the mix, it became more complex, because we almost always had to factor the numerator and the denominator in order for the terms to cancel. The first example we used is in the picture below.

As you can see, once you factor both expressions, any factors that they have in common will cancel. It is important to remember, however, that you can't cancel any terms until each term is connected by multiplication. If one of the terms has addition or subtraction in it, such as (x-2), you can only cancel the full (x-2) if both expressions have it in common. You can't cancel just the x or just the -2.

After everyone understood the idea of simplifying rational expressions (with variables), we moved on to also finding the domain and zeros of the simplified expression. We had worked with domain before, when we were talking about graphing earlier in the year, but just as a review we defined it again.

Domain: The set of all x values that work for the expression (in order to have a valid output).
An example of an invalid output would be any x values that would make the denominator 0, because then the expression would be "undefined".

We have not worked with zeros before, so this was a new definition for us.

Zeros: All x values that make a fraction equal 0.
This means any x value that will make the numerator 0, because any fraction with a numerator of 0 is equal to 0.

In this example, the two expressions simplified to (x+1)/(x-1). This means that the number that would make the denominator 0 is 1, which is why it is excluded from the domain. All other numbers work with x-1. The zero, or the number that makes the numerator 0, was -1, because the numerator was x+1, and if you add 1 to -1, the difference is 0.

We spent the rest of the class working on examples and finding the domain and zeros of different rational expressions. This was the gateway into Wednesday's class.


We started the class on Wednesday by going over the homework, which was all either simplifying fractions with variables or finding the domain and zero. One that people had a particular amount of trouble with was #31. Mainly the problem was factoring each of the expressions, because once they were factored the problem was actually fairly simple.

The denominator of the fraction could be factored using the difference of squares pattern that we learned earlier: a^2-b^2=(a-b)(a+b). Eventually, part of the numerator could be factored using the same pattern. Once everything was completely factored, everything in the denominator factored out, and only x+y remained in the numerator. That meant that the simplified answer was (x+y)/1, which is equal to just x+y.

On Wednesday, we expanded on the idea of simplifying these expressions by adding multiplication and division to the mix. In order to refresh our memory, Lisa gave us an example of a multiplication problem with two fractions without any variables. She showed us that you could simplify the fraction after you multiply, or you could simplify the fractions before multiplying in order to save yourself time and work.

Dividing fractions is very similar to multiplying fractions, because you can turn those problems into multiplication problems. The process is KCF, or Keep Change Flip. You keep the first fraction in your equation, change your sign from division to multiplication, and then flip your second fraction.

Since we all understood the basic principles of multiplying and dividing fractions, we quickly moved on to adding variables into the fractions. The process is similar, but you just need to factor the numerators and denominators of each fraction. That way you can cancel the terms before you do the actual multiplication, like in the second example in the picture above. A term that is in the numerator and the denominator of the same fraction will cancel, or a term that is in the numerator of one fraction and the denominator of another. A term that is on the same side of the fractions will never cancel. Once everything is cancelled, you can multiply to find your answer. I found a video on Youtube that explains multiplication of "algebraic fractions."

 Hope this clarified any questions you had!

- Molly Brock