Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section: Chapter Questions
Problem 7DE
Related questions
Question
![### Determine the Slant Asymptote for the Rational Function
To find the slant asymptote for the rational function \( f(x) = \frac{x^2 + 6x + 7}{x + 1} \), you need to perform polynomial long division.
#### Steps:
1. **Set Up the Division**:
- Dividend: \( x^2 + 6x + 7 \)
- Divisor: \( x + 1 \)
2. **Divide the Leading Terms**:
- Divide the leading term of the dividend (\( x^2 \)) by the leading term of the divisor (\( x \)).
\[
\frac{x^2}{x} = x
\]
3. **Multiply and Subtract**:
- Multiply \( x \) by the divisor (\( x + 1 \)):
\[
x \cdot (x + 1) = x^2 + x
\]
- Subtract this result from the dividend:
\[
(x^2 + 6x + 7) - (x^2 + x) = 5x + 7
\]
4. **Repeat the Process**:
- Divide the new leading term (\( 5x \)) by \( x \):
\[
\frac{5x}{x} = 5
\]
- Multiply \( 5 \) by the divisor (\( x + 1 \)):
\[
5 \cdot (x + 1) = 5x + 5
\]
- Subtract this result from \( 5x + 7 \):
\[
(5x + 7) - (5x + 5) = 2
\]
5. **Conclusion**:
- The quotient \( x + 5 \) is the equation of the slant asymptote.
- The remainder \( 2 \) does not affect the asymptote.
So, the slant asymptote is:
\[
y = x + 5
\]
This detailed explanation and the use of the Equation Editor Tool allow you to clearly show your work and ensure accuracy in finding the slant asymptote.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee316dc0-9fcf-44c0-a9c4-799945d7ab8e%2F84da78e0-2280-43bc-88c5-97046d747b48%2F8ajosd7_processed.png&w=3840&q=75)
Transcribed Image Text:### Determine the Slant Asymptote for the Rational Function
To find the slant asymptote for the rational function \( f(x) = \frac{x^2 + 6x + 7}{x + 1} \), you need to perform polynomial long division.
#### Steps:
1. **Set Up the Division**:
- Dividend: \( x^2 + 6x + 7 \)
- Divisor: \( x + 1 \)
2. **Divide the Leading Terms**:
- Divide the leading term of the dividend (\( x^2 \)) by the leading term of the divisor (\( x \)).
\[
\frac{x^2}{x} = x
\]
3. **Multiply and Subtract**:
- Multiply \( x \) by the divisor (\( x + 1 \)):
\[
x \cdot (x + 1) = x^2 + x
\]
- Subtract this result from the dividend:
\[
(x^2 + 6x + 7) - (x^2 + x) = 5x + 7
\]
4. **Repeat the Process**:
- Divide the new leading term (\( 5x \)) by \( x \):
\[
\frac{5x}{x} = 5
\]
- Multiply \( 5 \) by the divisor (\( x + 1 \)):
\[
5 \cdot (x + 1) = 5x + 5
\]
- Subtract this result from \( 5x + 7 \):
\[
(5x + 7) - (5x + 5) = 2
\]
5. **Conclusion**:
- The quotient \( x + 5 \) is the equation of the slant asymptote.
- The remainder \( 2 \) does not affect the asymptote.
So, the slant asymptote is:
\[
y = x + 5
\]
This detailed explanation and the use of the Equation Editor Tool allow you to clearly show your work and ensure accuracy in finding the slant asymptote.
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