How can rational expression be simplified

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 canceled 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 first do the division and then we will factor and cancel as much as we can. Here are the general formulas. In this case the least common denominator is So we need to get the denominators of these two fractions to a This is easy to do. In the first case we need to multiply the denominator by 2 to get 12 so we will multiply the numerator and denominator 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 finding the least common denominator for rational expressions. Here is it. This is 6. So, the least common denominator for this set of rational expression is. So, we simply need to multiply each term by an appropriate quantity to get this in the denominator and then do the addition and subtraction.

In this case there are only two factors and they both occur to the first power and so the least common denominator is. The final step is to do any multiplication in the numerator and simplify that up as much as possible. So we will write both of those down and then take the highest power for each. Here is the least common denominator for this rational expression.

In the last term recall that we need to do the multiplication prior to distributing the 3 through the parenthesis. Here is the simplification work for this part. The twist now is that you are looking for factors that are common to both the numerator and the denominator of the rational expression. Note that the other restriction is still explicitly part of the final expression.

Free Algebra Solver Error : Please Click on "Not a robot", then try downloading again. Worksheet and Answer key on simplifying rational expressions. Steps to simplify rational expressions.

Whenever you have terms added together, there are understood parentheses around them, like this:. You can only cancel off factors that is, entire expressions contained within parentheses , not terms that is, not just part of the contents of a pair of parentheses. To go inside the parentheses and try to cancel off part of the contents is like ripping off arms and legs of the poor little polynomial trapped inside. It'll be bleeding and oozing and flopping around on the floor, whimpering plaintively while sadly gazing up at you with big brown eyes Well, okay; maybe not.

But trying to cancel off only a portion of a factor would be like trying to do this:. Of course not. And if the above "cancellation" is illegitimate, then so also is this one:.

You can only cancel factors, not terms! You can use the Mathway widget below to practice simplifying a rational expression. Try the entered exercise, or type in your own exercise. Just like a fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator, a rational expression is simplified if it has no common factors, other than 1, in its numerator and denominator.

A rational expression is considered simplified if there are no common factors in its numerator and denominator. We use the Equivalent Fractions Property to simplify numerical fractions.

We restate it here as we will also use it to simplify rational expression s. If a , b , and c are numbers where , then and. Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see clearly stated. Every time we write a rational expression, we should make a similar statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.

Notice that the fraction is simplified because there are no more common factors. Throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, and. Did you notice that these are the same steps we took when we divided monomials in Polynomials?

To simplify rational expressions we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property. Be very careful as you remove common factors.

Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum. We now summarize the steps you should follow to simplify rational expressions. Usually, we leave the simplified rational expression in factored form. This way it is easy to check that we have removed all the common factors! Can you tell which values of x must be excluded in this example?

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors.

We know this fraction simplifies to. We also recognize that the numerator and denominator are opposites.