Simplifying Rational Expressions
Learning Outcomes
- Recognize and define a rational expression
- Determine the domain of a rational expression
- Simplify a rational expression
Try It
[ohm_question]189257[/ohm_question]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 [latex]x = 2[/latex]: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}=\frac{2}{0}\end{array}[/latex]
Remember that you can't divide by zero, so this means that for the expression [latex]\frac{x}{x-2}[/latex], [latex]x[/latex] cannot be [latex]2[/latex] because it will result in an undefined ratio. This means that for the expression [latex]\frac{x}{x-2}[/latex], [latex]x[/latex] cannot be [latex]2[/latex] 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:
[latex]x[/latex] is all real numbers where [latex]x\neq{a}[/latex]
Example
Identify the domain of the expression. [latex] \frac{3x+2}{x-4}[/latex]Answer: Find any values for x that would make the denominator equal [latex]0[/latex].
[latex]x–4=0[/latex]
When [latex]x=4[/latex], the denominator is equal to [latex]0[/latex].[latex]x=4[/latex]
Answer
The domain is all real numbers, except [latex]4[/latex].Check
You found that [latex]x\neq4[/latex]. 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 [latex]x=4[/latex], the numerator evaluates to [latex]14[/latex], but the denominator evaluates to [latex]0[/latex]. And since division by [latex]0[/latex] is undefined, this must be an excluded value.Example
Identify the domain of the expression. [latex] \frac{x+7}{{{x}^{2}}+8x-9}[/latex]Answer: Find any values for [latex]x[/latex] that would make the denominator equal to [latex]0[/latex] by setting the denominator equal to [latex]0[/latex] and solving the equation.
[latex]x^{2}+8x-9=0[/latex]
Solve the equation by factoring. The solutions are the values that are excluded from the domain.[latex]\begin{array}{c}(x+9)(x-1)=0\\x=-9\,\,\,\text{or}\,\,\,x=1\end{array}[/latex]
The domain is all real numbers except [latex]−9[/latex] and [latex]1[/latex].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:[latex]\frac{x^2+x-2}{x-1}[/latex]
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: [latex]2[/latex] and [latex]10[/latex] are factors of [latex]20[/latex], as are [latex]4, 5, 1, 20[/latex]. Terms are single numbers, or variables and numbers connected by multiplication. [latex]-4, 6x[/latex] 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 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: [latex]\dfrac{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-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 [latex]\frac{2}{2}=1\text{ and }\frac{x}{x}=1[/latex], so our expression simplifies to [latex]\dfrac{x}{6}[/latex]. 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 [latex]\frac{5x^{2}}{25x}[/latex].Answer: Rewrite the numerator and denominator as factors.
[latex]\frac{5x^{2}}{25x}=\frac{5\cdot{x}\cdot{x}}{5\cdot5\cdot{x}}[/latex]
Identify fractions that equal [latex]1[/latex], and then simplify.[latex]\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}[/latex]
Simplify.[latex] \frac{x}{5}[/latex]
Answer
[latex-display] \frac{5{{x}^{2}}}{25x}=\frac{x}{5}[/latex-display]Example
Simplify and state the domain for the expression. [latex] \frac{x+3}{{{x}^{2}}+12x+27}[/latex]Answer: To find the domain (and the excluded values), find the values where the denominator is equal to [latex]0[/latex]. Factor the quadratic and apply the zero product principle.
[latex]\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}[/latex]
The domain is all real numbers except [latex]x=-3[/latex] or [latex]x=-9[/latex]. Factor the numerator and denominator. Identify the factors that are the same in the numerator and denominator then simplify.[latex]\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]
[latex-display] \frac{x+3}{{{x}^{2}}+12x+27}=\frac{1}{x+9}[/latex-display] The domain is all real numbers except [latex]−3[/latex] and [latex]−9[/latex].Example
Simplify and state the domain for the expression. [latex]\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}[/latex]Answer: To find the domain, determine the values where the denominator is equal to [latex]0[/latex].
[latex]\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}[/latex]
The domain is all real numbers except [latex]0, 5[/latex], and [latex]−4[/latex]. 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.[latex] \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}[/latex]
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 [latex] \frac{x+6}{{{x}^{2}}-5x}[/latex-display] The domain is all real numbers except [latex]0, 5[/latex], and [latex]−4[/latex].Example
Simplify [latex]\frac{{x}^{2}-9}{{x}^{2}+4x+3}[/latex] and state the domain.Answer: To find the domain, determine the values where the denominator is equal to [latex]0[/latex]. 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 [latex]-3[/latex] and [latex]−1[/latex]. Now factor and simplify the entire rational expression. Notice the numerator is a difference of squares.
[latex] \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]
[latex-display]\frac{{x}^{2}-9}{{x}^{2}+4x+3}=\frac{x - 3}{x+1}[/latex-display] Domain: [latex]x\ne-3,-1[/latex]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 [latex]0[/latex].
- 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 [latex]0[/latex] is undefined, any values of the variable that result in a denominator of [latex]0[/latex] 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 [latex]1[/latex].Contribute!
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- Screenshot: Breaking Math. Provided by: Lumen Learning License: CC BY: Attribution.
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- Simplify and Give the Domain of Rational Expressions. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Unit 15: Rational Expressions, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution. License terms: Download for free at : http://cnx.org/contents/[email protected]:1/Preface.