Subsection 12.2.1 Simplifying Rational Expressions
Consider the two rational functions below. At first glance, which function looks simpler?
It can be argued that the function is simpler, at least with regard to the ease with which we can determine its domain, quickly evaluate it, and also determine where its function value is zero. All of these things are considerably more difficult with the function
These two functions are actually the
same function. Using factoring and the same process of canceling that’s used with numerical ratios, we will learn how to simplify the function
into the function
(The full process for simplifying
will be shown in
Example 8.)
To see a simple example of the process for simplifying a rational function or expression, let’s look at simplifying and by canceling common factors:
The statement “for ” was added when the factors of were canceled. This is because was undefined for so the simplified version must also be undefined for
Warning 12.2.2. Cancel Factors, not Terms.
It may be tempting to want to try to simplify into by canceling each that appears. But these ’s are terms (pieces that are added with other pieces), not factors. Canceling (an act of division) is only possible with factors (an act of multiplication).
The process of canceling factors is key to simplifying rational expressions. If the expression is not given in factored form, then this will be our first step. We’ll now look at a few more examples.
Example 12.2.3.
Simplify the rational function formula and state the domain of
Explanation.
To start, we’ll factor the numerator and denominator. We’ll then cancel any factors common to both the numerator and denominator.
The domain of this function will incorporate the explicit domain restriction that was stated when the factor of was canceled from both the numerator and denominator. We will also exclude from the domain as this value would make the denominator zero. Thus the domain of is
Warning 12.2.4.
When simplifying the function
in
Example 3, we cannot simply write
The reason is that this would result in our simplified version of the function
having a different domain than the original
More specifically, for our original function
it held that
was undefined, and this still needs to be true for the simplified form of
Example 12.2.5.
Simplify the rational function formula and state the domain of
Explanation.
The domain of this function will incorporate the explicit restrictions that were stated when the factors of and were canceled from both the numerator and denominator. Since the factor is still in the denominator, we also need the restriction that Therefore the domain of is
Example 12.2.6.
Simplify the expression
Explanation.
To start, we need to recognize that is not written in standard form (where terms are written from highest degree to lowest degree). Before attempting to factor this expression, we’ll re-write it as
Example 12.2.7.
Simplify the expression
Explanation.
To begin simplifying this expression, we will rewrite each polynomial in descending order. Then we’ll factor out the GCF, including the constant from both the numerator and denominator because their leading terms are negative.
Example 12.2.8.
Simplify the rational function formula and state the domain of
Explanation.
To simplify this rational function, we’ll first note that both the numerator and denominator have four terms. To factor them we’ll need to use factoring by grouping. (Note that if this technique didn’t work, very few other approaches would be possible.) Once we’ve used factoring by grouping, we’ll cancel any factors common to both the numerator and denominator and state the associated restrictions.
In determining the domain of this function, we’ll need to account for any implicit and explicit restrictions. When the factor was canceled, the explicit statement of was given. The denominator in the final simplified form of this function has There is no value of for which so the only restriction is that Therefore the domain is
Example 12.2.9.
Simplify the expression In this example, there are two variables. It is still possible that in examples like this, there can be domain restrictions when simplifying rational expressions. However since we are not studying functions of more than one variable, this textbook ignores domain restrictions with examples like this one.
Subsection 12.2.2 Multiplication of Rational Functions and Expressions
Recall the [cross-reference to target(s) "fact-multiplication-of-fractions" missing or not unique]
, which states that the product of two fractions is equal to the product of their numerators divided by the product of their denominators. We will use this same rule for multiplying rational expressions.
When multiplying fractions, one approach is to multiply the numerator and denominator, and then simplify the fraction that results by determining the greatest common factor in both the numerator and denominator, like this:
This approach works great when we can easily identify that is the greatest common factor in both and But in more complicated instances, it isn’t always an easy approach. It also won’t work particularly well when we have instead of as a factor, as we’ll see shortly.
Another approach to multiplying and simplifying fractions involves utilizing the prime factorization of each the numerator and denominator, like this:
The method for multiplying and simplifying rational expressions is nearly identical, as shown here:
This process will be used for both multiplying and dividing rational expressions. The main distinctions in various examples will be in the factoring methods required.
Example 12.2.10.
Multiply the rational expressions:
Explanation.
Note that to factor the second rational expression, we’ll want to re-write the terms in descending order for both the numerator and denominator. In the denominator, we’ll first factor out as the leading term is
Example 12.2.11.
Multiply the rational expressions: Note this book ignores domain restrictions on multivariable expressions.
Explanation.
We won’t need to factor anything in this example, and can simply multiply across and then simplify.
Subsection 12.2.3 Division of Rational Functions and Expressions
We can divide rational expressions using the [cross-reference to target(s) "fact-division-of-fractions" missing or not unique]
, which simply requires that we change dividing by an expression to multiplying by its reciprocal. Let’s look at a few examples.
Example 12.2.12.
Divide the rational expressions:
Remark 12.2.13.
In the first step of
Example 12, the restriction
was used. We hadn’t canceled anything yet, so why is there this restriction already? It’s because the original expression
had
in a denominator, which means that
is not a valid input. In the first step of simplifying, the
denominator went to the numerator and we lost the information that
was not a valid input, so we stated it explicitly. Always be sure to compare the restrictions of the original expression with each step throughout the process.
Example 12.2.14.
Simplify the rational expression using division:
Explanation.
To begin, we’ll note that the larger fraction bar is denoting division, so we will use multiplication by the reciprocal. After that, we’ll factor each expression and cancel any common factors.
Example 12.2.15.
Divide the rational expressions:
Example 12.2.16.
Divide the rational expressions:
Explanation.
Note here that we didn’t have to include a restriction in the very first step. That restriction would have been but since still cannot be inputted into any of the subsequent expressions, we don’t need to explicitly state as a restriction because the expressions tell us that implicitly already.
Example 12.2.17.
Divide the rational expressions: Note this book ignores domain restrictions on multivariable expressions.
Example 12.2.18.
Divide the rational expressions: Note this book ignores domain restrictions on multivariable expressions.