Module 8: Rational Expressions, Functions, and Equations
8.2 – Operations on Rational Expressions
Learning Objectives
(8.2.1) – Multiply and divide rational expressions
Multiply rational expressions
Divide rational expressions
(8.2.2) – Add and subtract rational expressions
Identify the least common denominator of two rational expressions
Add rational expressions
Subtract rational expressions
(8.2.3) – Complex rational expressions
(8.2.1) – Multiply and Divide Rational Expressions
Just as you can multiply and divide fractions, you can multiply and divide rational expressions. In fact, you use the same processes for multiplying and dividing rational expressions as you use for multiplying and dividing numeric fractions. The process is the same even though the expressions look different!
Multiply and Divide
Multiply Rational Expressions
Remember that there are two ways to multiply numeric fractions.
One way is to multiply the numerators and the denominators and then simplify the product, as shown here.
Notice that both methods result in the same product. In some cases you may find it easier to multiply and then simplify, while in others it may make more sense to simplify fractions before multiplying.
The same two approaches can be applied to rational expressions. Our first two examples apply both techniques to one expression. After that we will let you decide which works best for you.
Okay, that worked. But this time let’s simplify first, then multiply. When using this method, it helps to look for the greatest common factor. You can factor out any common factors, but finding the greatest one will take fewer steps.
Also, remember that when working with rational expressions, you should get into the habit of identifying any values for the variables that would result in division by 0. These excluded values must be eliminated from the domain, the set of all possible values of the variable. In the example above, [latex]\displaystyle \frac{5{{a}^{2}}}{14}\cdot \frac{7}{10{{a}^{3}}}[/latex], the domain is all real numbers where a is not equal to 0. When [latex]a=0[/latex], the denominator of the fraction [latex]\displaystyle \frac{7}{10a^{3}}[/latex] equals 0, which will make the fraction undefined.
Some rational expressions contain quadratic expressions and other multi-term polynomials. To multiply these rational expressions, the best approach is to first factor the polynomials and then look for common factors. (Multiplying the terms before factoring will often create complicated polynomials…and then you will have to factor these polynomials anyway! For this reason, it is easier to factor, simplify, and then multiply.) Just take it step by step, like in the examples below.
Note that in the answer above, you cannot simplify the rational expression any further. It may be tempting to express the 5’s in the numerator and denominator as the fraction [latex]\displaystyle \frac{5}{5}[/latex], but these 5’s are terms because they are being added or subtracted. Remember that only common factors, not terms, can be regrouped to form factors of 1!
In the following video we present another example of multiplying rational expressions.
Divide Rational Expressions
You’ve seen that you multiply rational expressions as you multiply numeric fractions. It should come as no surprise that you also divide rational expressions the same way you divide numeric fractions. Specifically, to divide rational expressions, keep the first rational expression, change the division sign to multiplication, and then take the reciprocal of the second rational expression.
Let’s begin by recalling division of numerical fractions.
Use the same process to divide rational expressions. You can think of division as multiplication by the reciprocal, and then use what you know about multiplication to simplify.
Reciprocal Architecture
You do still need to think about the domain, specifically the variable values that would make either denominator equal zero. But there’s a new consideration this time—because you divide by multiplying by the reciprocal of one of the rational expressions, you also need to find the values that would make the numerator of that expression equal zero. Have a look.
Knowing how to find the domain may seem unimportant here, but it will help you when you learn how to solve rational equations. To divide, multiply by the reciprocal.
Example
State the domain, then divide. [latex]\displaystyle \frac{5x^{2}}{9}\div\frac{15x^{3}}{27}[/latex]
Show Solution
State the Domain:
Find excluded values. 9 and 27 can never equal 0.
Because [latex]15x^{3}[/latex] becomes the denominator in the reciprocal of [latex] \displaystyle \frac{15{{x}^{3}}}{27}[/latex], you must find the values of x that would make [latex]15x^{3}[/latex] equal 0.
[latex]\begin{array}{c}15x^{3}=0\\x=0\,\text{is an excluded value}.\end{array}[/latex]
Divide:
State the quotient in simplest form. Rewrite division as multiplication by the reciprocal.
The domain is all real numbers except 0, [latex]−2[/latex], and [latex]−3[/latex].
Notice that once you rewrite the division as multiplication by a reciprocal, you follow the same process you used to multiply rational expressions.
In the video that follows, we present another example of dividing rational expressions.
(8.2.2) – Add and Subtract Rational Expressions
In beginning math, students usually learn how to add and subtract whole numbers before they are taught multiplication and division. However, with fractions and rational expressions, multiplication and division are sometimes taught first because these operations are easier to perform than addition and subtraction. Addition and subtraction of rational expressions are not as easy to perform as multiplication because, as with numeric fractions, the process involves finding common denominators.
Add and Subtract
Identify the least common denominator of two rational expressions
Add Rational Expressions
To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, given the rational expressions
[latex]\displaystyle \large\frac{6}{\left(x+3\right)\left(x+4\right)},\text{ and }\frac{9x}{\left(x+4\right)\left(x+5\right)}[/latex]
The LCD would be [latex]\left(x+3\right)\left(x+4\right)\left(x+5\right)[/latex].
To find the LCD, we count the greatest number of times a factor appears in each denominator, and make sure it is represented in the LCD that many times.
For example, in [latex]\displaystyle \large\frac{6}{\left(x+3\right)\left(x+4\right)}[/latex], [latex]\left(x+3\right)[/latex] is represented once and [latex]\left(x+4\right)[/latex] is represented once, so they both appear exactly once in the LCD.
In [latex]\displaystyle \large\frac{9x}{\left(x+4\right)\left(x+5\right)}[/latex], [latex]\left(x+4\right)[/latex] appears once, and [latex]\left(x+5\right)[/latex] appears once.
We have already accounted for [latex]\left(x+4\right)[/latex], so the LCD just needs one factor of [latex]\left(x+5\right)[/latex] to be complete.
Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD.
What do we mean by ” the form of 1″?
[latex]\displaystyle \frac{x+5}{x+5}=1[/latex] so multiplying an expression by it will not change it’s value.
For example, we would need to multiply the expression [latex]\displaystyle \frac{6}{\left(x+3\right)\left(x+4\right)}[/latex] by [latex]\displaystyle \frac{x+5}{x+5}[/latex] and the expression [latex]\displaystyle \frac{9x}{\left(x+4\right)\left(x+5\right)}[/latex] by [latex]\displaystyle \frac{x+3}{x+3}[/latex].
Hopefully this process will become clear after you practice it yourself. As you look through the examples on this page, try to identify the LCD before you look at the answers. Also, try figuring out which “form of 1” you will need to multiply each expression by so that it has the LCD.
Example
Add the rational expressions: [latex]\displaystyle \frac{5}{x}+\frac{6}{y}[/latex], and define the domain.
State the sum in simplest form.
Show Solution
First, let’s define the domain of each term. Since we have [latex]x[/latex] and [latex]y[/latex] in the denominators, we can say [latex]x\ne0 ,\text{ and }y\ne0[/latex].
Now we have to find the LCD. Since [latex]x[/latex] appears once and [latex]y[/latex] appears once, the LCD will be [latex]xy[/latex]. We then multiply each expression by the appropriate form of 1 to obtain [latex]xy[/latex] as the denominator for each fraction.
Now that the expressions have the same denominator, we simply add the numerators to find the sum.
[latex]\displaystyle \frac{6x+5y}{xy}[/latex]
The domain is [latex]x\ne-4[/latex]
Answer
[latex] \displaystyle \frac{2{{x}^{2}}}{x+4}+\frac{8x}{x+4}=2x,x\ne -4[/latex], [latex]x\ne0 ,\text{ and }y\ne0[/latex]
Analysis of the Solution
Multiplying by [latex]\displaystyle \frac{y}{y}[/latex] or [latex]\displaystyle \frac{x}{x}[/latex] does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression.
Here is one more example of adding rational expressions, but in this case, the expressions have denominators with multi-term polynomials. First, we will factor, then find the LCD. Note that [latex]x^2-4[/latex] is a difference of squares and can be factored using special products.
Example
Simplify[latex]\displaystyle \frac{2{{x}^{2}}}{{{x}^{2}}-4}+\frac{x}{x-2}[/latex], and give the domain.
State the result in simplest form.
Show Solution
Find the least common multiple by factoring each denominator. Multiply each factor the maximum number of times it appears in a single factorization. Remember that [latex]x[/latex] cannot be [latex]2[/latex] or [latex]-2[/latex] because the denominators would be 0.
[latex]\left(x+2\right)[/latex] appears a maximum of one time, as does [latex]\left(x–2\right)[/latex]. This means the LCM is [latex]\left(x+2\right)\left(x–2\right)[/latex].
Rewrite the original problem with the common denominator. It makes sense to keep the denominator in factored form in order to check for common factors.
Check for simplest form. Since neither [latex]\left(x+2\right)[/latex] nor [latex]\left(x-2\right)[/latex] is a factor of [latex]3{{x}^{2}}+2x[/latex], this expression is in simplest form.
In the video that follows, we present an example of adding two rational expression whose denominators are binomials with no common factors.
Subtract Rational Expressions
To subtract rational expressions, follow the same process you use to add rational expressions. You will need to be careful with signs, though.
Example
Subtract[latex]\displaystyle \frac{2}{t+1}-\frac{t-2}{{{t}^{2}}-t-2}[/latex], define the domain.
State the difference in simplest form.
Show Solution
Find the LCD of each expression. [latex]t+1[/latex] cannot be factored any further, but [latex]{{t}^{2}}-t-2[/latex] can be. Note that t cannot be [latex]-1[/latex] or [latex]2[/latex] because the denominators would be 0.
Find the least common multiple. [latex]t+1[/latex] appears exactly once in both of the expressions, so it will appear once in the least common denominator. [latex]t–2[/latex] also appears once.
This means that [latex]\left(t-2\right)\left(t+1\right)[/latex] is the least common multiple. In this case, it is easier to leave the common multiple in terms of the factors, so you will not multiply it out.
Use the least common multiple for your new common denominator, it will be the LCD.
Compare each original denominator and the new common denominator. Now rewrite the rational expressions to each have the common denominator of [latex]\left(t+1\right)\left(t–2\right)[/latex].
You need to multiply [latex]t+1[/latex] by [latex]t–2[/latex] to get the LCD, so multiply the entire rational expression by [latex] \displaystyle \frac{t-2}{t-2}[/latex].
The second expression already has a denominator of [latex]\left(t+1\right)\left(t–2\right)[/latex], so you do not need to multiply it by anything.
Subtract the numerators and simplify. Remember that parentheses need to be included around the second [latex]\left(t–2\right)[/latex] in the numerator because the whole quantity is subtracted. Otherwise you would be subtracting just the t.
In the next example, we will give less instruction. See if you can find the LCD yourself before you look at the answer.
Example
Subtract the rational expressions: [latex]\displaystyle \frac{6}{{x}^{2}+4x+4}-\frac{2}{{x}^{2}-4}[/latex], and define the domain.
State the difference in simplest form.
Show Solution
Note that the denominator of the first expression is a perfect square trinomial, and the denominator of the second expression is a difference of squares so they can be factored using special products.
In the last example, the LCD was [latex]\displaystyle \left(x+2\right)^2\left(x-2\right)[/latex]. The reason we need to include [latex]\left(x+2\right)[/latex] two times is because it appears two times in the expression [latex]\displaystyle \frac{6}{{x}^{2}+4x+4}[/latex].
The video that follows contains an example of subtracting rational expressions.
Try It
(8.2.3) – Complex Rational Expressions
Fractions and rational expressions can be interpreted as quotients. When both the dividend (numerator) and divisor (denominator) include fractions or rational expressions, you have something more complex than usual. Don’t fear—you have all the tools you need to simplify these quotients!
A complex fraction is the quotient of two fractions. These complex fractions are never considered to be in simplest form, but they can always be simplified using division of fractions. Remember, to divide fractions, you multiply by the reciprocal.
Before you multiply the numbers, it’s often helpful to factor the numbers. You can then use the factors to create a fraction equal to 1.
First combine the numerator and denominator by adding or subtracting, you may need to find a common denominator first. Note that we do not show the steps for finding a common denominator – so please review that if you are confused.
In the following video we will show a couple more examples of how to simplify complex fractions.
A complex rational expression is a quotient with rational expressions in the dividend, divisor, or in both. Simplify these in the exact same way as you would a complex fraction.
Rewrite the division as multiplication, using the reciprocal of the divisor. Note that the excluded values for this are [latex]-4[/latex], [latex]4[/latex] and [latex]5[/latex], because those values make the denominators of one of the fractions zero.
Factor the numerator and denominator, looking for common factors. In this case, [latex]x+5[/latex] and [latex]x–4[/latex] are common factors of the numerator and denominator. Notice that [latex] \frac{(x+5)(x-4)}{(x+5)(x-4)}[/latex] is equal to 1.
In the next video example we will show that simplifying a complex fraction may require factoring first.
The same ideas can be used when simplifying complex rational expressions that include more than one rational expression in the numerator or denominator. However, there is a shortcut that can be used. Compare these two examples of simplifying a complex fraction.
Combine the expressions in the numerator and denominator. To do this, rewrite the expressions using a common denominator. There is an excluded value of 0 because this makes the denominators of the fractions zero.
Rewrite the complex rational expression as a division problem. (When you are comfortable with the step of rewriting the complex rational fraction as a division problem, you might skip this step and go straight to rewriting it as multiplication.)
Factor the numerator and denominator, looking for common factors. In this case, [latex]x+3[/latex] and [latex]x^{2}[/latex] are common factors. We can now see there are two additional excluded values, [latex]-2[/latex] and [latex]-3[/latex].
Before combining the expressions, find a common denominator for all of the rational expressions. (In this case, [latex]x^{2}[/latex] is a common denominator.) Multiply by 1 in the form of a fraction with the common denominator in both numerator and denominator. (In this case, multiply by [latex]\displaystyle \frac{{{x}^{2}}}{{{x}^{2}}}[/latex].) There is an excluded value of 0 because this makes the denominators of the fractions zero.
Notice that the expression is no longer complex! You can simplify by factoring and identifying common factors. We can now see there are two additional excluded values, [latex]-2[/latex] and [latex]-3[/latex].
You may find the second method easier to use, but do try both ways to see what you prefer.
In our last example, we show a similar example as the one above.
Remember… an additional consideration for rational expressions is to determine what values are excluded from the domain. Since division by 0 is undefined, any values of the variables that result in a denominator of 0 must be excluded. Excluded values must be identified in the original equation, not from its factored form.Rational expressions are fractions containing polynomials. They can be simplified much like numeric fractions. To simplify a rational expression, first determine common factors of the numerator and denominator, and then remove them by rewriting them as expressions equal to 1.
Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution
Simplify and Give the Domain of Rational Expressions. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/tJiz5rEktBs. License: CC BY: Attribution
Screenshot: Multiply and Divide. Provided by: Lumen Learning. License: CC BY: Attribution
Multiply Rational Expressions and Give the Domain. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/Hj6gF1SNttk. License: CC BY: Attribution
Subtract Rational Expressions with Unlike Denominators and Give the Domain. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/MMlNtCrkakI. License: CC BY: Attribution
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Unit 15: Rational Expressions, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education. Located at: http://nrocnetwork.org/dm-opentext. License: CC BY: Attribution