Find the difference. See Examples 6–7. 2x3 + 7x2 – 3x + 7 6x3 – 4x2 – 9x – 3 -
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
![### Example Problem: Finding the Difference of Polynomials
**Problem Statement:**
Find the difference. See Examples 6–7.
\[
\begin{array}{r}
2x^3 + 7x^2 - 3x + 7 \\
- (6x^3 + 4x^2 + 9x + 3) \\
\hline
\end{array}
\]
To solve this problem, follow these steps:
1. **Identify and organize the polynomials:**
- The first polynomial is \(2x^3 + 7x^2 - 3x + 7\).
- The second polynomial to be subtracted is \(6x^3 + 4x^2 + 9x + 3\).
2. **Rewrite the second polynomial with a negative sign:**
```
- (6x^3 + 4x^2 + 9x + 3)
= -6x^3 - 4x^2 - 9x - 3
```
3. **Combine like terms:**
```
\begin{array}{r}
2x^3 + 7x^2 - 3x + 7 \\
-6x^3 - 4x^2 - 9x - 3 \\
\hline
\end{array}
```
4. **Compute the difference for each term:**
- For the \(x^3\) term: \(2x^3 - 6x^3 = -4x^3\)
- For the \(x^2\) term: \(7x^2 - 4x^2 = 3x^2\)
- For the \(x\) term: \(-3x - 9x = -12x\)
- For the constant term: \(7 - 3 = 4\)
5. **Write the final expression:**
\[
-4x^3 + 3x^2 - 12x + 4
\]
### Detailed Explanation:
In this problem, you are required to subtract the second polynomial from the first. To do so, you negate each term in the second polynomial and then add the resulting terms to the terms of the first polynomial. Combining like terms involves adding or](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c02ecae-9f3c-4011-90ce-878cb1c02c8e%2F123ea045-f1b3-4b6f-b45f-6bfcd03adbf5%2F4fi3bz2h_processed.png&w=3840&q=75)
Transcribed Image Text:### Example Problem: Finding the Difference of Polynomials
**Problem Statement:**
Find the difference. See Examples 6–7.
\[
\begin{array}{r}
2x^3 + 7x^2 - 3x + 7 \\
- (6x^3 + 4x^2 + 9x + 3) \\
\hline
\end{array}
\]
To solve this problem, follow these steps:
1. **Identify and organize the polynomials:**
- The first polynomial is \(2x^3 + 7x^2 - 3x + 7\).
- The second polynomial to be subtracted is \(6x^3 + 4x^2 + 9x + 3\).
2. **Rewrite the second polynomial with a negative sign:**
```
- (6x^3 + 4x^2 + 9x + 3)
= -6x^3 - 4x^2 - 9x - 3
```
3. **Combine like terms:**
```
\begin{array}{r}
2x^3 + 7x^2 - 3x + 7 \\
-6x^3 - 4x^2 - 9x - 3 \\
\hline
\end{array}
```
4. **Compute the difference for each term:**
- For the \(x^3\) term: \(2x^3 - 6x^3 = -4x^3\)
- For the \(x^2\) term: \(7x^2 - 4x^2 = 3x^2\)
- For the \(x\) term: \(-3x - 9x = -12x\)
- For the constant term: \(7 - 3 = 4\)
5. **Write the final expression:**
\[
-4x^3 + 3x^2 - 12x + 4
\]
### Detailed Explanation:
In this problem, you are required to subtract the second polynomial from the first. To do so, you negate each term in the second polynomial and then add the resulting terms to the terms of the first polynomial. Combining like terms involves adding or
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