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# Simplifying Rational Expressions

### Learning Outcomes

• Recognize and define a rational expression
• Determine the domain of a rational expression
• Simplify a rational expression
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 $\frac{4x^3}{12x^2}$ 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.

### Try It

[ohm_question]189257[/ohm_question]
Keep Calm and be Rational

## 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  $\frac{x}{x-2}$ for $x=2$

Substitute $x=2$

$\begin{array}{l}\frac{2}{2-2}=\frac{2}{0}\end{array}$

Remember that you can't divide by zero, so this means that for the expression $\frac{x}{x-2}$, $x$ cannot be  $2$ because it will result in an undefined ratio.  This means that for the expression $\frac{x}{x-2}$, $x$ 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.

### Domain of a rational expression or equation

The domain of a rational expression or equation is a collection of the values for the variable that will not result in an undefined mathematical operation such as division by zero.  For a = any real number, we can notate the domain in the following way:

$x$ is all real numbers where $x\neq{a}$

The reason you cannot divide any number c by zero $\left( \frac{c}{0}\,\,=\,\,? \right)$ is that you would have to find a number that when you multiply it by $0$ you would get back $c \left( ?\,\,\cdot \,\,0\,\,=\,\,c \right)$. There are no numbers that can do this, so we say “division by zero is undefined”. When 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 are called excluded values. Discard them right at the start before you go any further. (Note that although the denominator cannot be equal 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 where the 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. For rational expressions, the domain will exclude values for which the value of the denominator is $0$. The following two examples illustrate finding the domain of an expression.

### Example

Identify the domain of the expression. $\frac{3x+2}{x-4}$

Answer: Find any values for x that would make the denominator equal $0$.

$x–4=0$

When $x=4$, the denominator is equal to $0$.

$x=4$

The domain is all real numbers, except $4$.

#### Check

You found that $x\neq4$. Substitute that value into the expression to check that it gives an undefined mathematical operation. [latex-display]\begin{array}{c}\frac{3x+2}{x-4}\\\\\frac{3(4)+2}{(4)-4}\\\\\frac{12+2}{0}\\\\\frac{14}{0}\end{array}[/latex-display] You find that when $x=4$, the numerator evaluates to $14$, but the denominator evaluates to $0$. And since division by $0$ is undefined, this must be an excluded value.

In the next example we will identify the domain of a rational expression that contains a trinomial in the denominator.

### Example

Identify the domain of the expression. $\frac{x+7}{{{x}^{2}}+8x-9}$

Answer: Find any values for $x$ that would make the denominator equal to $0$ by setting the denominator equal to $0$ and solving the equation.

$x^{2}+8x-9=0$

Solve the equation by factoring. The solutions are the values that are excluded from the domain.

$\begin{array}{c}(x+9)(x-1)=0\\x=-9\,\,\,\text{or}\,\,\,x=1\end{array}$

The domain is all real numbers except $−9$ and $1$.

### Try It

[ohm_question]74947[/ohm_question]

## Simplify Rational Expressions

As with many other mathematical expressions and equations, it can be very helpful to simplify rational expressions.  We simplified rational expressions with monomial terms in the exponents module. Here we will combine what we know about factoring polynomials with factoring rational expressions that have monomial terms. The goal is to be able to simplify an expression such as this:

$\frac{x^2+x-2}{x-1}$

But before we dive in to simplifying rational expressions like the one above, let us review the difference between a factor,  a term,  and an expression.  This will hopefully help you avoid some common mistakes 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 $x^2$ are all terms. Expressions are groups of terms connected by addition and subtraction. $2x^2-5$ is an expression. This distinction is important when you are required to divide. Let us use an example to show why this is important.  The idea is that a number or variable divided by itself is equal to one, so we can factor a rational expression and identify common factors between the numerator and denominator. Simplify: $\dfrac{2x^2}{12x}$ The numerator and denominator of this fraction consist of factors. To simplify it, we can divide without being impeded by addition or subtraction. [latex-display]\begin{array}{cc}\dfrac{2x^2}{12x}\\=\dfrac{2\cdot{x}\cdot{x}}{2\cdot3\cdot2\cdot{x}}\\=\dfrac{\cancel{2}\cdot{\cancel{x}}\cdot{x}}{\cancel{2}\cdot3\cdot2\cdot{\cancel{x}}}\end{array}[/latex-display] The common factors between the numerator and denominator are 2 and x, so we can "cancel" them to show that $\frac{2}{2}=1\text{ and }\frac{x}{x}=1$, so our expression simplifies to $\dfrac{x}{6}$. The next example provides a reminder of how to simplify a monomial with variables and exponents. We will then use this idea to simplify a rational expression and define it's domain.

### Example

Simplify $\frac{5x^{2}}{25x}$.

Answer: Rewrite the numerator and denominator as factors.

$\frac{5x^{2}}{25x}=\frac{5\cdot{x}\cdot{x}}{5\cdot5\cdot{x}}$

Identify fractions that equal $1$, and then simplify.

$\displaystyle\large\begin{array}{c}\frac{5\cdot{x}\cdot{x}}{5\cdot5\cdot{x}}\\\\=\frac{\cancel{5}\cdot{\cancel{x}}\cdot{x}}{\cancel{5}\cdot5\cdot{\cancel{x}}}\\\\=\frac{x}{5}\normalsize\cdot1\end{array}$

Simplify.

$\frac{x}{5}$

[latex-display] \frac{5{{x}^{2}}}{25x}=\frac{x}{5}[/latex-display]

We can summarize the process as follows: Factor the numerator, factor the denominator, identify factors that are common to the numerator and denominator, cancel them to represent division, and simplify. When simplifying rational expressions, it is a good habit to always consider the domain first. This will come in handy when you begin solving rational equations a bit later on. When finding the domain of an expression, you always start with the original expression because variable terms may be factored out as part of the simplification process. Compare that to the expression $\dfrac{2x^2+x}{12-2x}$. 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 cannot do that, it will break math!
Be careful not to break math when working with rational expressions.
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 the numerator and denominator. This turns expressions connected by addition and subtraction into terms connected by multiplication.  Once we have factored the numerator and denominator, the same principles of simplifying will once again apply.

### Example

Simplify and state the domain for the expression. $\frac{x+3}{{{x}^{2}}+12x+27}$

Answer: To find the domain (and the excluded values), find the values where the denominator is equal to $0$. Factor the quadratic and apply the zero product principle.

$\begin{array}{c}x+3=0\,\,\,\,\,\,\text{or}\,\,\,\,\,\,x+9=0\\x=0-3\,\,\,\,\,\,\text{or}\,\,\,\,\,\,x=0-9\\x=-3\,\,\,\,\,\,\text{or}\,\,\,\,\,\,x=-9\end{array}$

The domain is all real numbers except $x=-3$ or $x=-9$. Factor the numerator and denominator.  Identify the factors that are the same in the numerator and denominator then simplify.

$\large\begin{array}{c}\frac{x+3}{x^{2}+12x+27}\\\\=\frac{x+3}{\left(x+3\right)\left(x+9\right)}\\\\\frac{\cancel{x+3}}{\cancel{\left(x+3\right)}\left(x+9\right)}\\\\\normalsize=1\cdot\dfrac{1}{x+9}\end{array}$

[latex-display] \frac{x+3}{{{x}^{2}}+12x+27}=\frac{1}{x+9}[/latex-display] The domain is all real numbers except $−3$ and $−9$.

### Example

Simplify and state the domain for the expression. $\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}$

Answer: To find the domain, determine the values where the denominator is equal to $0$.

$\begin{array}{r}x^{3}-x^{2}-20x=0\\x\left(x^{2}-x-20\right)=0\\x\left(x-5\right)\left(x+4\right)=0\end{array}$

The domain is all real numbers except $0, 5$, and $−4$. To simplify, factor the numerator and denominator of the rational expression. Identify the factors that are the same in the numerator and denominator then simplify.

$\large\begin{array}{c}\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}\\\\=\frac{\left(x+4\right)\left(x+6\right)}{x\left(x-5\right)\left(x+4\right)}\\\\=\frac{\cancel{\left(x+4\right)}\left(x+6\right)}{x\left(x-5\right)\cancel{\left(x+4\right)}}\end{array}$

It is acceptable to either leave the denominator in factored form or to distribute/multiply. [latex-display] \frac{x+6}{x(x-5)}[/latex] or $\frac{x+6}{{{x}^{2}}-5x}[/latex-display] The domain is all real numbers except [latex]0, 5$, and $−4$.

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

### Example

Simplify $\frac{{x}^{2}-9}{{x}^{2}+4x+3}$ and state the domain.

Answer: To find the domain, determine the values where the denominator is equal to $0$. Be sure to factor the denominator first. [latex-display]\left(x+3\right)\left(x+1\right)=0[/latex-display] The domain is all real numbers except $-3$ and $−1$. Now factor and simplify the entire rational expression. Notice the numerator is a difference of squares.

$\large\begin{array}{c}\frac{{x}^{2}-9}{{x}^{2}+4x+3}\\\\=\frac{\left(x+3\right)\left(x-3\right)}{\left(x+3\right)\left(x+1\right)}\\\\=\frac{\cancel{\left(x+3\right)}\left(x-3\right)}{\cancel{\left(x+3\right)}\left(x+1\right)}\end{array}$

[latex-display]\frac{{x}^{2}-9}{{x}^{2}+4x+3}=\frac{x - 3}{x+1}[/latex-display] Domain: $x\ne-3,-1$

In the following video, we present additional examples of simplifying and finding the domain of a rational expression. [embed]https://youtu.be/tJiz5rEktBs[/embed]

### Steps for Simplifying a Rational Expression

To simplify a rational expression, follow these steps:
• Determine the domain. The excluded values are those values for the variable that result in the expression having a denominator of $0$.
• Factor the numerator and denominator.
• Cancel out common factors in the numerator and denominator and simplify.

### Try It

[ohm_question]3343[/ohm_question]

## 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 variable that result in a denominator of $0$ must be excluded from the domain. 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 $1$.

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