12.7: Excluded Values for Rational Expressions
What if you had a rational expression like
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CK-12 Foundation: 1207S Rational Expressions
Watch this video for more examples of how to simplify rational expressions.
YourTeacher: Simplifying Rational Expressions
Guidance
A simplified rational expression is one where the numerator and denominator have no common factors. In order to simplify an expression to lowest terms , we factor the numerator and denominator as much as we can and cancel common factors from the numerator and the denominator.
Simplify Rational Expressions
Example A
Reduce each rational expression to simplest terms.
a)
b)
c)
Solution
a)
b)
c)
When reducing fractions, you are only allowed to cancel common
factors
from the denominator but NOT common terms. For example, in the expression
Why can’t we do that? When we cross out terms that are part of a sum or a difference, we’re violating the order of operations (PEMDAS). Remember, the fraction bar means division. When we perform the operation
Using numbers instead of variables makes it more obvious that canceling individual terms doesn’t work. You can see that
Find Excluded Values of Rational Expressions
Whenever there’s a variable expression in the denominator of a fraction, we must remember that the denominator could be zero when the independent variable takes on certain values. Those values, corresponding to the vertical asymptotes of the function, are called excluded values. To find the excluded values, we simply set the denominator equal to zero and solve the resulting equation.
Example B
Find the excluded values of the following expressions.
a)
b)
Solution
a) \begin{align*}\text{When we set the denominator equal to zero we obtain:} \quad \ \ x+4=0 \Rightarrow x=-4\!\\ \\ \text{So} \ \mathbf{-4} \ \text{is the excluded value.}\end{align*}
b) @$$\begin{align*}\text{When we set the denominator equal to zero we obtain:} \qquad x^2-x-6=0\!\\ \\ \text{Solve by factoring:} \qquad \qquad \qquad \qquad \qquad \qquad \ \qquad \qquad \qquad (x-3)(x+2)=0\!\\ \\ {\;} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \qquad \qquad \Rightarrow x=3 \ \text{and}\ x = -2\!\\ \\ \text{So}\ \mathbf{3}\ \mathbf{and}\ \mathbf{-2} \ \text{are the excluded values.}\end{align*}@$$
Removable Zeros
Removable zeros are those zeros from the original expression, but is not a zero for the simplified version of the expression. However, we have to keep track of them, because they were zeros in the original expression. This is illustrated in the following examples.
Example C
Determine the removable values of @$\begin{align*}\frac{4x-2}{2x^2+x-1}\end{align*}@$ .
Solution:
Notice that in the expressions in Example A, we removed a division by zero when we simplified the problem. For instance, we rewrote @$\begin{align*}\frac{4x-2}{2x^2+x-1}\end{align*}@$ as @$\begin{align*}\frac{2(2x-1)}{(2x-1)(x+1)}\end{align*}@$ . The denominator of this expression is zero when @$\begin{align*}x = \frac{1}{2}\end{align*}@$ or when @$\begin{align*}x = -1\end{align*}@$ .
However, when we cancel common factors, we simplify the expression to @$\begin{align*}\frac{2}{x+1}\end{align*}@$ . This reduced form allows the value @$\begin{align*}x = \frac{1}{2}\end{align*}@$ , so @$\begin{align*}x = -1\end{align*}@$ is its only excluded value.
Technically the original expression and the simplified expression are not the same. When we reduce a radical expression to its simplest form, we should specify the removed excluded value. In other words, we should write our final answer as @$\begin{align*}\frac{4x-2}{2x^2+x-1}=\frac{2}{x+1}, x \neq \frac{1}{2}\end{align*}@$ .
Example D
Determine the removable values of the expressions from Example A parts b and c.
Solution:
We should write the answer from Example A, part b as @$\begin{align*}\frac{x^2-2x+1}{8x-8}=\frac{x-1}{8}, x \neq 1\end{align*}@$ .
The answer from Example A, part c as @$\begin{align*}\frac{x^2-4}{x^2-5x+6}=\frac{x+2}{x-3}, x \neq 2\end{align*}@$ .
Watch this video for help with the Examples above.
CK-12 Foundation: Rational Expressions
Vocabulary
- Whenever there’s a variable expression in the denominator of a fraction, we must remember that the denominator could be zero when the independent variable takes on certain values. Those values, corresponding to the vertical asymptotes of the function, are called excluded values.
- Removable zeros are those zeros from the original expression, but is not a zero for the simplified version of the expression.
Guided Practice
Find the excluded values of @$\begin{align*}\frac{4}{x^2-5x}\end{align*}@$ .
Solution
@$$\begin{align*}\text{When we set the denominator equal to zero we obtain:} \quad \ \ x^2-5x=0\!\\ \\ \text{Solve by factoring:} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ x(x-5)=0\!\\ \\ {\;} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ \Rightarrow x=0 \ \text{and} \ x = 5\!\\ \\ \text{So} \ \mathbf{0 \ and \ 5}\ \text{are the excluded values.}\end{align*}@$$
Explore More
Reduce each fraction to lowest terms.
- @$\begin{align*}\frac{4}{2x-8}\end{align*}@$
- @$\begin{align*}\frac{x^2+2x}{x}\end{align*}@$
- @$\begin{align*}\frac{9x+3}{12x+4}\end{align*}@$
- @$\begin{align*}\frac{6x^2+2x}{4x}\end{align*}@$
- @$\begin{align*}\frac{x-2}{x^2-4x+4}\end{align*}@$
- @$\begin{align*}\frac{x^2-9}{5x+15}\end{align*}@$
- @$\begin{align*}\frac{x^2+6x+8}{x^2+4x}\end{align*}@$
- @$\begin{align*}\frac{2x^2+10x}{x^2+10x+25}\end{align*}@$
- @$\begin{align*}\frac{x^2+6x+5}{x^2-x-2}\end{align*}@$
- @$\begin{align*}\frac{x^2-16}{x^2+2x-8}\end{align*}@$
- @$\begin{align*}\frac{3x^2+3x-18}{2x^2+5x-3}\end{align*}@$
- @$\begin{align*}\frac{x^3+x^2-20x}{6x^2+6x-120}\end{align*}@$
Find the excluded values for each rational expression.
- @$\begin{align*}\frac{2}{x}\end{align*}@$
- @$\begin{align*}\frac{4}{x+2}\end{align*}@$
- @$\begin{align*}\frac{2x-1}{(x-1)^2}\end{align*}@$
- @$\begin{align*}\frac{3x+1}{x^2-4}\end{align*}@$
- @$\begin{align*}\frac{x^2}{x^2+9}\end{align*}@$
- @$\begin{align*}\frac{2x^2+3x-1}{x^2-3x-28}\end{align*}@$
- @$\begin{align*}\frac{5x^3-4}{x^2+3x}\end{align*}@$
- @$\begin{align*}\frac{9}{x^3+11x^2+30x}\end{align*}@$
- @$\begin{align*}\frac{4x-1}{x^2+3x-5}\end{align*}@$
- @$\begin{align*}\frac{5x+11}{3x^2-2x-4}\end{align*}@$
- @$\begin{align*}\frac{x^2-1}{2x^2+x+3}\end{align*}@$
- @$\begin{align*}\frac{12}{x^2+6x+1}\end{align*}@$
- In an electrical circuit with resistors placed in parallel, the reciprocal of the total resistance is equal to the sum of the reciprocals of each resistance. @$\begin{align*}\frac{1}{R_c}=\frac{1}{R_1}+\frac{1}{R_2}\end{align*}@$ . If @$\begin{align*}R_1 = 25 \ \Omega\end{align*}@$ and the total resistance is @$\begin{align*}R_c = 10 \ \Omega\end{align*}@$ , what is the resistance @$\begin{align*}R_2\end{align*}@$ ?
- Suppose that two objects attract each other with a gravitational force of 20 Newtons. If the distance between the two objects is doubled, what is the new force of attraction between the two objects?
- Suppose that two objects attract each other with a gravitational force of 36 Newtons. If the mass of both objects was doubled, and if the distance between the objects was doubled, then what would be the new force of attraction between the two objects?
- A sphere with radius @$\begin{align*}R\end{align*}@$ has a volume of @$\begin{align*}\frac{4}{3} \pi R^3\end{align*}@$ and a surface area of @$\begin{align*}4 \pi R^2\end{align*}@$ . Find the ratio the surface area to the volume of a sphere.
- The side of a cube is increased by a factor of 2. Find the ratio of the old volume to the new volume.
- The radius of a sphere is decreased by 4 units. Find the ratio of the old volume to the new volume.
Oblique Asymptote
An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division.Rational Expression
A rational expression is a fraction with polynomials in the numerator and the denominator.Vertical Asymptote
A vertical asymptote is a vertical line marking a specific value toward which the graph of a function may approach, but will never reach.Image Attributions
Description
Learning Objectives
Here you'll learn how to reduce rational expressions to their simplest terms. You'll also learn how to find the excluded values of rational expressions.
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Date Created:
Oct 01, 2012Last Modified:
Jul 31, 2015Vocabulary
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