Excluded Values for Rational Expressions

Excluding values that result in division by zero

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Excluded Values for Rational Expressions

You are given two rational expressions\begin{align*}\frac{x^2-10x+25}{x^2-1}\end{align*}  and \begin{align*}\frac{x^2-4}{x^3-8}\end{align*} . Can these expresisions be reduced further?

Simplify Rational Expressions

You have gained experience working with rational functions so far. In this section, you will continue simplifying rational expressions by factoring.

To simplify a rational expression means to reduce the fraction into its lowest terms.

To do this, you will need to remember a property about multiplication.

For all real values \begin{align*}a \text{ and } b\end{align*}, and \begin{align*}b \neq 0, \frac{ab}{b}=a\end{align*}.

Let's simplify the following rational expression:

\begin{align*}\frac{x^2-2x+1}{8x-8}\end{align*}

Factor both pieces of the rational expression and reduce.

\begin{align*}& \frac{x^2-2x+1}{8x-8} \rightarrow \frac{(x-1)(x-1)}{8(x-1)}\\ & \frac{x^2-2x+1}{8x-8}=\frac{x-1}{8}\end{align*}

Finding Excluded Values of Rational Expressions

As stated in a previous Concept, excluded values are also called points of discontinuity. These are the values that make the denominator equal to zero and are not part of the domain.

Find the excluded values of \begin{align*}\frac{2x+1}{x^2-x-6}\end{align*}.

Factor the denominator of the rational expression.

\begin{align*}\frac{2x+1}{x^2-x-6}=\frac{2x+1}{(x+2)(x-3)}\end{align*}

Find the value that makes each factor equal zero.

\begin{align*}x=-2, x=3\end{align*}

These are excluded values of the domain of the rational expression.

Let's use simplifying rational expressions to solve the following real-world application:

The gravitational force between two objects in given by the formula \begin{align*}F=\frac{G(m_1m_2)}{(d^2)}\end{align*}. The gravitation constant is given by \begin{align*}G=6.67 \times 10^{-11} (N \cdot m^2/kg^2)\end{align*}. The force of attraction between the Earth and the Moon is \begin{align*}F=2.0 \times 10^{20} \ N\end{align*} (with masses of \begin{align*}m_1=5.97 \times 10^{24} \ kg\end{align*} for the Earth and \begin{align*}m_2=7.36 \times 10^{22} \ kg\end{align*} for the Moon).

What is the distance between the Earth and the Moon?

\begin{align*}&\text{Let's start with the Law of Gravitation formula}. && F =G\frac {m_1m_2}{d^2} \\ &\text{Now plug in the known values}. && 2.0\times 10^{20} N =6.67\times 10^{-11} \frac {N \cdot m^2}{kg^2}.\frac {(5.97\times 10^{24}kg)(7.36\times 10^{22}kg)}{d^2} \\ &\text{Multiply the masses together}. && 2.0\times 10^{20} N =6.67\times 10^{-11} \frac {N \cdot m^2}{kg^2}.\frac {4.39\times 10^{47}kg^2}{d^2} \\ &\text{Cancel the}\ kg^2\ \text{units}. && 2.0\times 10^{20} N = 6.67 \times 10^{-11} \frac{N \cdot m^2} {\cancel{kg^2}} \cdot \frac{4.39 \times 10^{47} \cancel{kg^2}} {d^2} \\ &\text{Multiply the numbers in the numerator}. && 2.0\times 10^{20} N \frac {2.93\times 10^{37}}{d^2}N \cdot m^2 \\ &\text{Multiply both sides by}\ d^2. && 2.0\times 10^{20} N \cdot d^2 =\frac {2.93\times 10^{37}}{d^2} \cdot d^2 \cdot N \cdot m^2 \\ &\text{Cancel common factors}. && 2.0\times 10^{20} N \cdot d^{2} = \frac{2.93 \times 10^{37}} {\cancel{d^2}} \cdot \cancel{d^2} \cdot N \cdot m^2 \\ &\text{Simplify}. && 2.0\times 10^{20} N \cdot d^2 = 2.93\times 10^{37}N \cdot m^2 \\ &\text{Divide both sides by}\ 2.0 \times 10^{20} N. && d^2 =\frac {2.93 \times 10^{37}}{2.0\times 10^{20}}\frac {N \cdot m^2}{N} \\ &\text{Simplify}. && d^2 =1.465\times 10^{17} m^2 \\ &\text{Take the square root of both sides}. && d =3.84 \times 10^8m\\\end{align*}

Examples

Earlier, you were asked if you could reduce \begin{align*}\frac{x^2-10x+25}{x^2-1}\end{align*} and \begin{align*}\frac{x^2-4}{x^3-8}\end{align*} further. We can determine this by factoring the expressions and seeing if anything cancels out.

Let's take a look at the first expression.

\begin{align*}& \frac{x^2-10x+25}{x^2-1} \rightarrow \frac{(x-5)(x-5)}{(x-1)(x+1)}\end{align*}As you can see, after factoring nothing cancels out - so it cannot be simplified further.

Now, let's take a look at the second expression.

\begin{align*}& \frac{x^2-4}{x^3-8} \rightarrow \frac{(x-2)(x+2)}{(x-2)(x^2+2x-4)}\\ &= \frac{(x+2)}{x^2+2x-4}\\\end{align*}

As you can see, after you factor you can cancel the \begin{align*}(x-2) \end{align*} term - leaving you with your simplified final answer.

Example 2

Find the excluded values by simplifying \begin{align*}\frac{4x-2}{2x^2+x-1}\end{align*}.

Both the numerator and denominator can be factored using methods learned in previous Concepts.

\begin{align*}\frac{4x-2}{2x^2+x-1} \rightarrow \frac{2(2x-1)}{(2x-1)(x+1)}\end{align*}

The expression \begin{align*}(2x-1)\end{align*} appears in both the numerator and denominator and can be canceled. The expression becomes:

\begin{align*}\frac{4x-2}{2x^2+x-1}=\frac{2}{x+1}\end{align*}

Since \begin{align*}x+1=0\Rightarrow x=-1\end{align*}, then \begin{align*}x=-1\end{align*} is an excluded value.

Review

Reduce each fraction to lowest terms.

1. \begin{align*}\frac{4}{2x-8}\end{align*}
2. \begin{align*}\frac{x^2+2x}{x}\end{align*}
3. \begin{align*}\frac{9x+3}{12x+4}\end{align*}
4. \begin{align*}\frac{6x^2+2x}{4x}\end{align*}
5. \begin{align*}\frac{x-2}{x^2-4x+4}\end{align*}
6. \begin{align*}\frac{x^2-9}{5x+15}\end{align*}
7. \begin{align*}\frac{x^2+6x+8}{x^2+4x}\end{align*}
8. \begin{align*}\frac{2x^2+10x}{x^2+10x+25}\end{align*}
9. \begin{align*}\frac{x^2+6x+5}{x^2-x-2}\end{align*}
10. \begin{align*}\frac{x^2-16}{x^2+2x-8}\end{align*}
11. \begin{align*}\frac{3x^2+3x-18}{2x^2+5x-3}\end{align*}
12. \begin{align*}\frac{x^3+x^2-20x}{6x^2+6x-120}\end{align*}

Find the excluded values for each rational expression.

1. \begin{align*}\frac{2}{x}\end{align*}
2. \begin{align*}\frac{4}{x+2}\end{align*}
3. \begin{align*}\frac{2x-1}{(x-1)^2}\end{align*}
4. \begin{align*}\frac{3x+1}{x^2-4}\end{align*}
5. \begin{align*}\frac{x^2}{x^2+9}\end{align*}
6. \begin{align*}\frac{2x^2+3x-1}{x^2-3x-28}\end{align*}
7. \begin{align*}\frac{5x^3-4}{x^2+3x}\end{align*}
8. \begin{align*}\frac{9}{x^3+11x^2+30x}\end{align*}
9. \begin{align*}\frac{4x-1}{x^2+3x-5}\end{align*}
10. \begin{align*}\frac{5x+11}{3x^2-2x-4}\end{align*}
11. \begin{align*}\frac{x^2-1}{2x^3+x+3}\end{align*}
12. \begin{align*}\frac{12}{x^2+6x+1}\end{align*}
13. 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*}?
14. 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?
15. 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?
16. 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 of the surface area to the volume of the sphere.
17. The side of a cube is increased by a factor of two. Find the ratio of the old volume to the new volume.
18. The radius of a sphere is decreased by four units. Find the ratio of the old volume to the new volume.

Mixed Review

1. Name \begin{align*}4p^6+7p^3-9\end{align*}.
2. Simplify \begin{align*}(4b^2+b+7b^3 )+(5b^2-6b^4+b^3)\end{align*}. Write the answer in standard form.
3. State the Zero Product Property.
4. Why can’t the Zero Product Property be used in this situation: \begin{align*}(5x+1)(x-4)=2\end{align*}?
5. Shelly earns $4.85 an hour plus$15 in tips. Graph her possible total earnings for one day of work.
6. Multiply and simplify: \begin{align*}(-4x^2+8x-1)(-7x^2+6x+8)\end{align*}.
7. A rectangle’s perimeter is 65 yards. The length is 7 more yards than its width.

To see the Review answers, open this PDF file and look for section 12.4.

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Color Highlighted Text Notes

Vocabulary Language: English

TermDefinition
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.