Consider the polynomial defined by p(x) = (x − a)(x − b) ... (x − z). a. How many additions and multiplications are involved in this calculation? Explain. b. Calculate p(7).
Consider the polynomial defined by p(x) = (x − a)(x − b) ... (x − z). a. How many additions and multiplications are involved in this calculation? Explain. b. Calculate p(7).
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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![**Polynomial Analysis - Educational Exercise**
Consider the polynomial defined by \( p(x) = (x - a)(x - b) \ldots (x - z) \).
### Questions:
1. **How many additions and multiplications are involved in this calculation? Explain.**
2. **Calculate \( p(7) \).**
### Explanation:
**Question 1:**
To answer how many additions and multiplications are involved in the calculation of \( p(x) \), it's essential to consider the form of the polynomial. Given the product form \( p(x) = (x - a)(x - b) \ldots (x - z) \):
- Each term \((x - a_j)\) involves **one subtraction** (which is a form of addition in arithmetic operations).
- There are \( n \) such terms if there are \( n \) roots \( a, b, \ldots, z \).
To multiply two terms together, it involves **one multiplication**.
Therefore, to compute the product:
- For \( n \) terms, there will be \( n-1 \) multiplicative operations needed to get the final polynomial's value at a specific \( x \).
**Question 2:**
To calculate \( p(7) \), substitute \( x = 7 \) into each term of the polynomial and compute the product. This step involves simple substitution followed by arithmetic operations. The details depend on the specific values of \( a, b, \ldots, z \) which are not given in this problem. Hence, the general form of the substitution would look like:
\[ p(7) = (7 - a)(7 - b) \ldots (7 - z) \]
This gives the value of the polynomial at \( x = 7 \).
### Note:
If specific values for \( a, b, \ldots, z \) were provided, we would include those values in the calculation for \( p(7) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe6e77107-ef1d-4e39-badf-4decafa47f1b%2Fc14f35e7-3543-4b5f-83df-aa9e6a20936d%2F96su4ov_processed.png&w=3840&q=75)
Transcribed Image Text:**Polynomial Analysis - Educational Exercise**
Consider the polynomial defined by \( p(x) = (x - a)(x - b) \ldots (x - z) \).
### Questions:
1. **How many additions and multiplications are involved in this calculation? Explain.**
2. **Calculate \( p(7) \).**
### Explanation:
**Question 1:**
To answer how many additions and multiplications are involved in the calculation of \( p(x) \), it's essential to consider the form of the polynomial. Given the product form \( p(x) = (x - a)(x - b) \ldots (x - z) \):
- Each term \((x - a_j)\) involves **one subtraction** (which is a form of addition in arithmetic operations).
- There are \( n \) such terms if there are \( n \) roots \( a, b, \ldots, z \).
To multiply two terms together, it involves **one multiplication**.
Therefore, to compute the product:
- For \( n \) terms, there will be \( n-1 \) multiplicative operations needed to get the final polynomial's value at a specific \( x \).
**Question 2:**
To calculate \( p(7) \), substitute \( x = 7 \) into each term of the polynomial and compute the product. This step involves simple substitution followed by arithmetic operations. The details depend on the specific values of \( a, b, \ldots, z \) which are not given in this problem. Hence, the general form of the substitution would look like:
\[ p(7) = (7 - a)(7 - b) \ldots (7 - z) \]
This gives the value of the polynomial at \( x = 7 \).
### Note:
If specific values for \( a, b, \ldots, z \) were provided, we would include those values in the calculation for \( p(7) \).
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