Use synthetic division to divide. (Simplify your answer completely.) (3x³ + 11x² + 15x − 4) + (x + 5) X = -5
Use synthetic division to divide. (Simplify your answer completely.) (3x³ + 11x² + 15x − 4) + (x + 5) X = -5
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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![**Topic: Polynomial Division Using Synthetic Division**
### Problem Statement:
Use synthetic division to divide. (Simplify your answer completely.)
\[ \frac{(3x^3 + 11x^2 + 15x - 4)}{(x + 5)} \]
\[ \text{where } x \neq -5 \]
**Note:** There is a placeholder box after which the solution steps and final simplified answer are to be written.
### Explanation:
#### 1. Identify the coefficients of the dividend polynomial:
\[ f(x) = 3x^3 + 11x^2 + 15x - 4 \]
The coefficients are: 3, 11, 15, and -4.
#### 2. Identify the root of the divisor \( (x + 5) \):
The root is \( x = -5 \).
#### 3. Set up the synthetic division:
- Write the root \( -5 \) on the left.
- Write the coefficients of \( f(x) \) in a row: 3, 11, 15, -4.
#### 4. Perform the synthetic division steps:
\[ \begin{array}{r|rrrr}
-5 & 3 & 11 & 15 & -4 \\
& & -15 & 20 & -175 \\
\hline
& 3 & -4 & 35 & -179 \\
\end{array} \]
Here's how the division is carried out:
- Bring down the first coefficient (3) directly.
- Multiply -5 by 3 and write the product (-15) under the next coefficient (11).
- Add the column: 11 - 15 = -4.
- Continue this process:
- Multiply -5 by -4 to get 20,
- Add the column: 15 + 20 = 35,
- Multiply -5 by 35 to get -175,
- Add the column: -4 - 175 = -179.
#### 5. Write the quotient:
The quotient polynomial is obtained from the resulting row (ignoring the remainder):
\[ 3x^2 - 4x + 35 \]
#### 6. Write the remainder:
The remainder is -179.
### Simplified Answer:
The result of the division is:
\[ \boxed{3x^2 - 4x +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb5ff0f4-c9f6-49ea-9161-fbb64a60fa05%2Fc52858c3-41ba-462d-a21b-5dd8d4bc3987%2Fkurrplj_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Polynomial Division Using Synthetic Division**
### Problem Statement:
Use synthetic division to divide. (Simplify your answer completely.)
\[ \frac{(3x^3 + 11x^2 + 15x - 4)}{(x + 5)} \]
\[ \text{where } x \neq -5 \]
**Note:** There is a placeholder box after which the solution steps and final simplified answer are to be written.
### Explanation:
#### 1. Identify the coefficients of the dividend polynomial:
\[ f(x) = 3x^3 + 11x^2 + 15x - 4 \]
The coefficients are: 3, 11, 15, and -4.
#### 2. Identify the root of the divisor \( (x + 5) \):
The root is \( x = -5 \).
#### 3. Set up the synthetic division:
- Write the root \( -5 \) on the left.
- Write the coefficients of \( f(x) \) in a row: 3, 11, 15, -4.
#### 4. Perform the synthetic division steps:
\[ \begin{array}{r|rrrr}
-5 & 3 & 11 & 15 & -4 \\
& & -15 & 20 & -175 \\
\hline
& 3 & -4 & 35 & -179 \\
\end{array} \]
Here's how the division is carried out:
- Bring down the first coefficient (3) directly.
- Multiply -5 by 3 and write the product (-15) under the next coefficient (11).
- Add the column: 11 - 15 = -4.
- Continue this process:
- Multiply -5 by -4 to get 20,
- Add the column: 15 + 20 = 35,
- Multiply -5 by 35 to get -175,
- Add the column: -4 - 175 = -179.
#### 5. Write the quotient:
The quotient polynomial is obtained from the resulting row (ignoring the remainder):
\[ 3x^2 - 4x + 35 \]
#### 6. Write the remainder:
The remainder is -179.
### Simplified Answer:
The result of the division is:
\[ \boxed{3x^2 - 4x +
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