Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Long Division of Polynomials
#### Problem Statement:
Use long division to divide and simplify the following expression completely:
\[ \frac{8x^2 - 37x - 15}{x - 5} \]
Given that \( x \neq 5 \), perform the division and simplify your answer.
#### Solution Process:
1. **Setup the Long Division:** Place \( 8x^2 - 37x - 15 \) (the dividend) inside the long division symbol and \( x - 5 \) (the divisor) outside.
2. **Divide the First Term:** Divide the leading term of the dividend by the leading term of the divisor:
\[ \frac{8x^2}{x} = 8x \]
3. **Multiply and Subtract:** Multiply \( 8x \) by \( x - 5 \) to get:
\[ 8x^2 - 40x \]
Subtract \( 8x^2 - 40x \) from \( 8x^2 - 37x - 15 \) to get:
\[ 8x^2 - 37x - 15 - (8x^2 - 40x) = 3x - 15 \]
4. **Repeat the Process:** Repeat, dividing the first term of the new polynomial by the leading term of the divisor:
\[ \frac{3x}{x} = 3 \]
5. **Multiply and Subtract:** Multiply \( 3 \) by \( x - 5 \):
\[ 3(x - 5) = 3x - 15 \]
Subtract to get:
\[ 3x - 15 - (3x - 15) = 0 \]
So, the final answer to the division problem is:
\[ 8x + 3 \]
Putting it all together, we obtain:
\[ \frac{8x^2 - 37x - 15}{x - 5} = 8x + 3 \quad \text{for} \quad x \neq 5 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2367e66-1ae4-4371-aa7b-db391deab840%2Fda3efb0a-2a90-4419-aea9-6e870ef8c360%2Fj0smdzi_processed.png&w=3840&q=75)
Transcribed Image Text:### Long Division of Polynomials
#### Problem Statement:
Use long division to divide and simplify the following expression completely:
\[ \frac{8x^2 - 37x - 15}{x - 5} \]
Given that \( x \neq 5 \), perform the division and simplify your answer.
#### Solution Process:
1. **Setup the Long Division:** Place \( 8x^2 - 37x - 15 \) (the dividend) inside the long division symbol and \( x - 5 \) (the divisor) outside.
2. **Divide the First Term:** Divide the leading term of the dividend by the leading term of the divisor:
\[ \frac{8x^2}{x} = 8x \]
3. **Multiply and Subtract:** Multiply \( 8x \) by \( x - 5 \) to get:
\[ 8x^2 - 40x \]
Subtract \( 8x^2 - 40x \) from \( 8x^2 - 37x - 15 \) to get:
\[ 8x^2 - 37x - 15 - (8x^2 - 40x) = 3x - 15 \]
4. **Repeat the Process:** Repeat, dividing the first term of the new polynomial by the leading term of the divisor:
\[ \frac{3x}{x} = 3 \]
5. **Multiply and Subtract:** Multiply \( 3 \) by \( x - 5 \):
\[ 3(x - 5) = 3x - 15 \]
Subtract to get:
\[ 3x - 15 - (3x - 15) = 0 \]
So, the final answer to the division problem is:
\[ 8x + 3 \]
Putting it all together, we obtain:
\[ \frac{8x^2 - 37x - 15}{x - 5} = 8x + 3 \quad \text{for} \quad x \neq 5 \]
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