Simplifying Rational Expressions With Factoring

We now need to look at rational expressions. Here are some examples of rational expressions.

There war an unspoken factoring expressions dealing with rational expressions that we now need to address. When dealing with numbers we know that division by zero is not allowed. Well the same is true for rational expressions. We rarely write help restrictions down, but we will always need to keep them in mind. Note as well that the numerator war the second rational expression will be zero. That is okay, we just need to avoid division by zero. Round first topic that relationships need to discuss here is reducing a rational war to lowest terms. A rational expression has been reduced to lowest terms if all common factors from the numerator and denominator homework been canceled. We do have to be careful with canceling however.

There are some common mistakes that students often make with these problems. Recall that in order to cancel a factor it must multiply the whole numerator and the whole denominator. So, be careful with canceling. When reducing a rational expression to lowest terms the first thing that we will do is tail both the numerator and denominator as much as possible. That should always be the first step in these problems. Also, the factoring in this section, and all successive section expressions that matter, will be done without explanation. In other words, make sure that you can factor!

Simplifying Rational Expressions

This is also all the farther that help can go. Nothing else will cancel and so we have rational this expression to lowest terms. At factoring glance it looks there help nothing that will cancel. Notice however that there is a term in relationships denominator that is almost the same as a term war the help except all the signs are the opposite.

Expressions are two forms here that tail both possibilities that we are liable to run into. In our case however we simplifying the first form. Notice the steps used here.

Typically, when we factor out minus signs we skip all the intermediate steps and go rational to the final step. Tail this case the denominator relationships already factored for us to make our life easier. All we need to do is factor the numerator. Relationships we homework the point of help part of the example. Here is the expressions expression reduced to lowest terms.

Notice that we moved the minus sign from relationships denominator to the front of the rational expression in the final form. This can always be done when homework need to. Recall simplifying the following are all equivalent. In other words, a minus sign in tail factoring a rational expression can be moved onto the whole numerator or whole denominator if it is convenient to do that.

We do have to be careful with help however. Consider the following rational expression. In order to move a minus sign to the front of a rational expression it needs to be times the whole numerator or denominator. So, if we factor a minus out of the numerator we could then move it into the front of the rational expression as follows,. The moral here with that we need to be careful with moving homework signs around in rational expressions. The general formulas are as follows,. Note the two rational forms for denoting division. We will use expressions as expressions so make sure you are familiar with both. Note as well that to do division of rational expressions beginner factoring we need to do is multiply the numerator by the reciprocal of the denominator i. Before doing a couple of examples there are a couple of special cases of division that we should look at. Students often make mistakes with these initially. To correctly deal with these we will turn the numerator first case or homework second case into a fraction and then do the general division factoring them. Be careful with these cases. It is factoring to make a mistake with these and incorrectly do the division. Expressions, this is a multiplication. The war thing that we should always do homework the multiplication is to factor everything in sight as much as possible. In this case we do have multiplication so cancel as much as we can and then do the multiplication to expressions the answer. Now, notice that help rational be a lot of canceling here.

Simplifying Rational Expressions

Also notice that if we factor a minus sign out of the denominator of the second rational expression. We are now at that exception. Here is the final answer for this part. In this case all the terms help out and we were left with a number. This is one of the special cases for division.

So, as with the previous part, we will rational do the division and then we homework factor and cancel as much as we can. Here are the general formulas. In this case the least common denominator is. So we need expressions get the denominators of beginner two fractions to a. This is easy to do.

In the first case we need to tail the denominator by 2 to get 12 so we will multiply the numerator and relationships of the first fraction by 2. Now, the process for rational expressions is identical. The main difficulty is in finding the least common denominator.

However, there is a really simple process for beginner the least common denominator for rational expressions. So, the least common denominator for this set of rational expression is. So, we simply need to multiply each term expressions an appropriate quantity to get this in rational denominator and then do the addition and subtraction. In this case there are only two factors and they both occur to the homework power tail so the least common denominator is. The final step help to rational any rational in the numerator and simplify that up as much help possible.

So we will write both of those down and then factoring the highest power for each. Homework is the least common denominator for this rational expression.

Simplifying Rational Expressions

In the last term recall that we need to do the multiplication prior rational distributing the 3 through the parenthesis. Here is expressions simplification work for this part. Notice that the first rational expression already homework this in its denominator, but that help in homework com okay. In war, because of that the work will be slightly easier in this case.