Simplify the rational expression by using long division or synthetic d x4 +9x³3x² 18x + 2 x²-2 | x² +2
Simplify the rational expression by using long division or synthetic d x4 +9x³3x² 18x + 2 x²-2 | x² +2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Simplification of Rational Expressions Using Long Division or Synthetic Division
**Objective:**
Simplify the given rational expression by using long division or synthetic division.
**Problem:**
Simplify the expression
\[
\frac{x^4 + 9x^3 - 3x^2 - 18x + 2}{x^2 - 2}
\]
subject to the condition: \( x^2 \neq 2 \).
**Steps:**
1. **Long Division:**
- Begin by dividing the highest degree term of the numerator by the highest degree term of the denominator.
- Subtract the result from the original polynomial.
- Repeat the process with the new, smaller polynomial until the remaining polynomial's degree is less than the denominator's degree.
2. **Synthetic Division:**
- Use the zeros of the polynomial in the denominator to perform synthetic division.
- Replace the polynomial division with repeated arithmetic operations.
**Interactive Component:**
Provide input for each step of long or synthetic division in the following interactive box, and check your answers as you go.
```interactive box showing steps```
Ensure you do not attempt division where the denominator is equal to zero (i.e., avoid \( x^2 = 2 \)) as this leads to undefined behavior in mathematics.
Use these methods to understand how to simplify complex rational expressions effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2367e66-1ae4-4371-aa7b-db391deab840%2Fea1bbac2-4b3e-41bc-bfea-08ddc99eddd0%2Fb5od02c_processed.png&w=3840&q=75)
Transcribed Image Text:### Simplification of Rational Expressions Using Long Division or Synthetic Division
**Objective:**
Simplify the given rational expression by using long division or synthetic division.
**Problem:**
Simplify the expression
\[
\frac{x^4 + 9x^3 - 3x^2 - 18x + 2}{x^2 - 2}
\]
subject to the condition: \( x^2 \neq 2 \).
**Steps:**
1. **Long Division:**
- Begin by dividing the highest degree term of the numerator by the highest degree term of the denominator.
- Subtract the result from the original polynomial.
- Repeat the process with the new, smaller polynomial until the remaining polynomial's degree is less than the denominator's degree.
2. **Synthetic Division:**
- Use the zeros of the polynomial in the denominator to perform synthetic division.
- Replace the polynomial division with repeated arithmetic operations.
**Interactive Component:**
Provide input for each step of long or synthetic division in the following interactive box, and check your answers as you go.
```interactive box showing steps```
Ensure you do not attempt division where the denominator is equal to zero (i.e., avoid \( x^2 = 2 \)) as this leads to undefined behavior in mathematics.
Use these methods to understand how to simplify complex rational expressions effectively.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
