Rational expressions are fractions that have a polynomial in the numerator, denominator, or both. Although rational expressions can seem complicated because they contain variables, they can be simplified using the techniques used to simplify expressions such as [latex]\frac{4x^3}{12x^2}[/latex] combined with techniques for factoring polynomials. There are a couple ways to get yourself into trouble when working with rational expressions, equations and functions.  One of them is dividing by zero, and the other is trying to divide across addition or subtraction.

Determine the domain of a rational expression

One sure way you can break math is to divide by zero. Consider the following rational expression evaluated at x = 2:

Evaluate  [latex]\frac{x}{x-2}[/latex] for [latex]x=2[/latex]

Substitute [latex]x=2[/latex]

[latex]\begin{array}{l}\frac{2}{2-2}\\\text{}\\=\frac{2}{0}\end{array}[/latex]

This means that for the expression [latex]\frac{x}{x-2}[/latex], x cannot be 2 because it will result in an undefined ratio. In general, finding values for a variable that will not result in division by zero is called finding the domain. Finding the domain of a rational expression or function will help you not break math.

The reason you cannot divide any number c by zero [latex] \left( \frac{c}{0}\,\,=\,\,? \right)\\[/latex] is that you would have to find a number that when you multiply it by 0 you would get back [latex]c \left( ?\,\,\cdot \,\,0\,\,=\,\,c \right)[/latex]. There are no numbers that can do this, so we say “division by zero is undefined”. In simplifying rational expressions you need to pay attention to what values of the variable(s) in the expression would make the denominator equal zero. These values cannot be included in the domain, so they’re called excluded values. Discard them right at the start, before you go any further.

(Note that although the denominator cannot be equivalent to 0, the numerator can—this is why you only look for excluded values in the denominator of a rational expression.)

For rational expressions, the domain will exclude values for which the value of the denominator is 0. The following example illustrates finding the domain of an expression. Note that this is exactly the same algebra used to find the domain of a function.

Simplify Rational Expressions

Before we dive in to simplifying rational expressions, let’s review the difference between a factor,  a term,  and an expression.  This will hopefully help you avoid another way to break math when you are simplifying rational expressions.

Factors are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number: 2 and 10 are factors of 20, as are 4, 5, 1, 20.

Terms are single numbers, or variables and numbers connected by multiplication. -4, 6x and [latex]x^2[/latex] are all terms.

Expressions are groups of terms connected by addition and subtraction.  [latex]2x^2-5[/latex] is an expression.

This distinction is important when you are required to divide.  Let’s use an example to show why this is important.

Simplify: [latex]\large\frac{2x^2}{12x}[/latex]

The numerator and denominator of this fraction consist of factors. To simplify it, we can divide without being impeded by addition or subtraction.

[latex]\begin{array}{cc}\large\frac{2x^2}{12x}\\=\large\frac{2\cdot{x}\cdot{x}}{2\cdot3\cdot2\cdot{x}}\\=\large\frac{\cancel{2}\cdot{\cancel{x}}\cdot{x}}{\cancel{2}\cdot3\cdot2\cdot{\cancel{x}}}\end{array}[/latex]

We can do this because [latex]\frac{2}{2}=1\text{ and }\frac{x}{x}=1[/latex], so our expression simplifies to [latex]\large\frac{x}{6}[/latex]

Compare that to the expression [latex]\large\frac{2x^2+x}{12-2x}[/latex], notice the denominator and numerator consist of two terms connected by addition and subtraction.  We have to tip-toe around the addition and subtraction.  When asked to simplify it is tempting to want to cancel out like terms as we did when we just had factors. But you can’t do that, it will break math!

Breaking Math

In the examples that follow, the numerator and the denominator are polynomials with more than one term, and we will show you how to properly simplify them by factoring – which turns expressions connected by addition and subtraction into terms connected by multiplication.

We will show one last example of simplifying a rational expression. See if you can recognize the special product in the numerator.

In the following video we present another example of finding the domain of a rational expression.

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.

[latex] \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}[/latex]

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

[latex]\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}[/latex]

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.

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, [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]\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]\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.

[latex]\frac{2}{3}\div\frac{5}{9}=\frac{2}{3}\cdot\frac{9}{5}=\frac{18}{15}=\frac{6}{5}[/latex]

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.

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.

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

Adding 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]\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]\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]\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]\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]\large\frac{6}{\left(x+3\right)\left(x+4\right)}[/latex] by [latex]\frac{x+5}{x+5}[/latex] and the expression [latex]\frac{9x}{\left(x+4\right)\left(x+5\right)}[/latex] by [latex]\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.

Analysis of the Solution

Multiplying by [latex]\frac{y}{y}[/latex] or [latex]\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.

In the video that follows, we present an example of adding two rational expression whose denominators are binomials with no common factors.

Subtracting 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.

In the next example, we will give less instruction.  See if you can find the LCD yourself before you look at the answer.

Analysis of the solution

In the last example, the LCD was  [latex]\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]\frac{6}{{x}^{2}+4x+4}[/latex].
The video that follows contains an example of subtracting rational expressions.

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.

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

Complex Rational Expressions

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.

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.

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.

Summary

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.

Rational expressions are multiplied and divided the same way as numeric fractions. To multiply, first find the greatest common factors of the numerator and denominator. Next, regroup the factors to make fractions equivalent to one. Then, multiply any remaining factors. To divide, first rewrite the division as multiplication by the reciprocal of the denominator. The steps are then the same as for multiplication.

When expressing a product or quotient, it is important to state the excluded values. These are all values of a variable that would make a denominator equal zero at any step in the calculations.

Complex rational expressions are quotients with rational expressions in the divisor, dividend, or both. When written in fractional form, they appear to be fractions within a fraction. These can be simplified by first treating the quotient as a division problem. Then you can rewrite the division as multiplication using the reciprocal of the divisor. Or you can simplify the complex rational expression by multiplying both the numerator and denominator by a denominator common to all rational expressions within the complex expression. This can help simplify the complex expression even faster.