Define and Simplify Rational Expressions
Learning Outcomes
- Define and simplify rational expressions
- Identify the domain of a rational expression
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]
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{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].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: [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. 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] We can do this because [latex]\frac{2}{2}=1\text{ and }\frac{x}{x}=1[/latex], so our expression simplifies to [latex]\dfrac{x}{6}[/latex]. Compare that to the expression [latex]\dfrac{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 cannot do that, it will break math!
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.