## 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
• 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.

$\displaystyle \frac{4}{5}\cdot \frac{9}{8}=\frac{36}{40}=\frac{3\cdot 3\cdot 2\cdot 2}{5\cdot 2\cdot 2\cdot 2}=\frac{3\cdot 3\cdot \cancel{2}\cdot\cancel{2}}{5\cdot \cancel{2}\cdot\cancel{2}\cdot 2}=\frac{3\cdot 3}{5\cdot 2}\cdot 1=\frac{9}{10}$

A second way is to factor and simplify the fractions before performing the multiplication.

$\displaystyle \frac{4}{5}\cdot\frac{9}{8}=\frac{2\cdot2}{5}\cdot\frac{3\cdot3}{2\cdot2\cdot2}=\frac{\cancel{2}\cdot\cancel{2}\cdot3\cdot3}{\cancel{2}\cdot5\cdot\cancel{2}\cdot2}=1\cdot\frac{3\cdot3}{5\cdot2}=\frac{9}{10}$

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.

### Example

Multiply.$\displaystyle \frac{5{{a}^{2}}}{14}\cdot \frac{7}{10{{a}^{3}}}$

State the product in simplest form.

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.

### Example

Multiply.  $\displaystyle \frac{5a^{2}}{14}\cdot\frac{7}{10a^{3}}$

State the product in simplest form.

Both methods produced the same answer.

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, $\displaystyle \frac{5{{a}^{2}}}{14}\cdot \frac{7}{10{{a}^{3}}}$, the domain is all real numbers where a is not equal to 0. When $a=0$, the denominator of the fraction $\displaystyle \frac{7}{10a^{3}}$ 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.

### Example

Multiply.  $\displaystyle \frac{{{a}^{2}}-a-2}{5a}\cdot \frac{10a}{a+1}\,\,,\,\,\,\,\,\,a\,\ne \,\,-1\,,\,\,0$

State the product in simplest form.

### Example

Multiply.  $\displaystyle \frac{a^{2}+4a+4}{2a^{2}-a-10}\cdot\frac{a+5}{a^{2}+2a},\,\,\,a\neq-2,0,\frac{5}{2}$

State the product in simplest form.

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 $\displaystyle \frac{5}{5}$, 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.

$\displaystyle \frac{2}{3}\div\frac{5}{9}=\frac{2}{3}\cdot\frac{9}{5}=\frac{18}{15}=\frac{6}{5}$

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.  $\displaystyle \frac{5x^{2}}{9}\div\frac{15x^{3}}{27}$

### Example

Divide.  $\displaystyle \frac{3x^{2}}{x+2}\div\frac{6x^{4}}{\left(x^{2}+5x+6\right)}$

State the quotient in simplest form, and express the domain of the expression.

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.

### Identify the least common denominator of two 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

$\displaystyle \large\frac{6}{\left(x+3\right)\left(x+4\right)},\text{ and }\frac{9x}{\left(x+4\right)\left(x+5\right)}$

The LCD would be $\left(x+3\right)\left(x+4\right)\left(x+5\right)$.

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 $\displaystyle \large\frac{6}{\left(x+3\right)\left(x+4\right)}$, $\left(x+3\right)$ is represented once and  $\left(x+4\right)$ is represented once, so they both appear exactly once in the LCD.

In $\displaystyle \large\frac{9x}{\left(x+4\right)\left(x+5\right)}$, $\left(x+4\right)$ appears once, and $\left(x+5\right)$ appears once.

We have already accounted for $\left(x+4\right)$, so the LCD just needs one factor of $\left(x+5\right)$ 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″?

$\displaystyle \frac{x+5}{x+5}=1$ so multiplying an expression by it will not change it’s value.

For example, we would need to multiply the expression $\displaystyle \frac{6}{\left(x+3\right)\left(x+4\right)}$ by $\displaystyle \frac{x+5}{x+5}$ and the expression $\displaystyle \frac{9x}{\left(x+4\right)\left(x+5\right)}$ by $\displaystyle \frac{x+3}{x+3}$.

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: $\displaystyle \frac{5}{x}+\frac{6}{y}$, and define the domain.

State the sum in simplest form.

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 $x^2-4$ is a difference of squares and can be factored using special products.

### Example

Simplify$\displaystyle \frac{2{{x}^{2}}}{{{x}^{2}}-4}+\frac{x}{x-2}$, and give the domain.

State the result 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$\displaystyle \frac{2}{t+1}-\frac{t-2}{{{t}^{2}}-t-2}$, define the domain.

State the difference in simplest form.

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: $\displaystyle \frac{6}{{x}^{2}+4x+4}-\frac{2}{{x}^{2}-4}$, and define the domain.

State the difference in simplest form.

The video that follows contains an example of subtracting rational expressions.

# (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.

### Example

Simplify.

$\displaystyle\large\frac{\,\frac{12}{35}\,}{\,\frac{6}{7}\,}$

If two fractions appear in the numerator or denominator (or both), first combine them. Then simplify the quotient as shown above.

### Example

Simplify.

$\displaystyle \frac{\,\frac{3}{4}+\frac{1}{2}\,}{\,\frac{4}{5}-\frac{1}{10}\,}$

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.

### Example

Simplify.

$\displaystyle \frac{\,\,\frac{x+5}{{{x}^{2}}-16}\,}{\,\,\frac{{{x}^{2}}-\,\,25}{x-4}\,}$

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.

### Example

Simplify.

$\displaystyle \frac{\,\,1-\frac{9}{{{x}^{2}}}\,\,}{\,\,1+\frac{5}{x}+\frac{6}{{{x}^{2}}}\,\,}$

### Example

Simplify.

$\displaystyle \frac{1-\frac{9}{{{x}^{2}}}}{1+\frac{5}{x}+\frac{6}{{{x}^{2}}}}$

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.