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学習ガイド > Intermediate Algebra

Define and Simplify Rational Expressions

Learning Outcomes

  • Define and simplify rational expressions
  • Identify the domain of 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 4x312x2\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.

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=2x = 2:

Evaluate  xx−2\frac{x}{x-2} for x=2x=2

Substitute x=2x=2

22−2=20\begin{array}{l}\frac{2}{2-2}=\frac{2}{0}\end{array}

This means that for the expression xx−2\frac{x}{x-2}, xx cannot be 22 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:

 xx is all real numbers where x≠ax\neq{a}

The reason you cannot divide any number c by zero (c0  =  ?) \left( \frac{c}{0}\,\,=\,\,? \right) is that you would have to find a number that when you multiply it by 00 you would get back c(?  ⋅  0  =  c)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 00, 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 00. 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.

Example

Identify the domain of the expression. x+7x2+8x−9 \frac{x+7}{{{x}^{2}}+8x-9}

Answer: Find any values for xx that would make the denominator equal to 00 by setting the denominator equal to 00 and solving the equation.

x2+8x−9=0x^{2}+8x-9=0

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

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

The domain is all real numbers except −9−9 and 11.

Simplify Rational Expressions

Before we dive in to simplifying rational expressions, let us 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: 22 and 1010 are factors of 2020, as are 4,5,1,204, 5, 1, 20. Terms are single numbers, or variables and numbers connected by multiplication. −4,6x-4, 6x and x2x^2 are all terms. Expressions are groups of terms connected by addition and subtraction. 2x2−52x^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. Simplify: 2x212x\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. 2x212x=2⋅x⋅x2⋅3⋅2⋅x=2⋅x⋅x2⋅3⋅2⋅x\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} We can do this because 22=1 and xx=1\frac{2}{2}=1\text{ and }\frac{x}{x}=1, so our expression simplifies to x6\dfrac{x}{6}. Compare that to the expression 2x2+x12−2x\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!
Shattered pottery strewn across the floor. 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. This turns expressions connected by addition and subtraction into terms connected by multiplication.

Example

Simplify and state the domain for the expression. x+3x2+12x+27 \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 00. Factor the quadratic and apply the zero product principle.

x+3=0      or      x+9=0x=0−3      or      x=0−9x=−3      or      x=−9\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=−3x=-3 or x=−9x=-9. Factor the numerator and denominator.  Identify the factors that are the same in the numerator and denominator then simplify.

x+3x2+12x+27=x+3(x+3)(x+9)x+3(x+3)(x+9)=1⋅1x+9\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}

x+3x2+12x+27=1x+9 \frac{x+3}{{{x}^{2}}+12x+27}=\frac{1}{x+9} The domain is all real numbers except −3−3 and −9−9.

Example

Simplify and state the domain for the expression. x2+10x+24x3−x2−20x\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}

Answer: To find the domain, determine the values where the denominator is equal to 00.

x3−x2−20x=0x(x2−x−20)=0x(x−5)(x+4)=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 0,50, 5, and −4−4. 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.

x2+10x+24x3−x2−20x=(x+4)(x+6)x(x−5)(x+4)=(x+4)(x+6)x(x−5)(x+4) \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. x+6x(x−5)[/latex]or[latex]x+6x2−5x \frac{x+6}{x(x-5)}[/latex] or [latex] \frac{x+6}{{{x}^{2}}-5x} The domain is all real numbers except 0,50, 5, and −4−4.

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

Example

Simplify x2−9x2+4x+3\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 00. Be sure to factor the denominator first. (x+3)(x+1)=0\left(x+3\right)\left(x+1\right)=0 The domain is all real numbers except −3-3 and −1−1. Now factor and simplify the entire rational expression. Notice the numerator is a difference of squares.

x2−9x2+4x+3=(x+3)(x−3)(x+3)(x+1)=(x+3)(x−3)(x+3)(x+1) \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}

x2−9x2+4x+3=x−3x+1\frac{{x}^{2}-9}{{x}^{2}+4x+3}=\frac{x - 3}{x+1} Domain: x≠−3,−1x\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 00.
  • Factor the numerator and denominator.
  • Cancel out common factors in the numerator and denominator and simplify.

Summary

An additional consideration for rational expressions is to determine what values are excluded from the domain. Since division by 00 is undefined, any values of the variable that result in a denominator of 00 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 then cancel them out.

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