Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
For the polynomial below, 1 is a zero.
![### Polynomial Factorization
**For the polynomial below, 1 is a zero.**
\[ f(x) = x^3 - 5x^2 + 3x + 1 \]
**Express \( f(x) \) as a product of linear factors.**
\[ f(x) = \]
In the image, you can also see an interactive element (probably from an educational platform) showing an icon with three options and the lowercase letter "i" along with typical icons for "ok," "reset," and "help." This likely indicates that the user is expected to input their answer or select different functionalities to assist in solving the problem.
### Explanation:
To solve this problem, follow these steps:
1. **Identify the given zero:** We are told that \(1\) is a zero of the polynomial.
2. **Polynomial Division:** Use synthetic or long division to divide the polynomial \( f(x) \) by \( x - 1 \).
3. **Factorization:** Express the quotient as a product of linear factors.
### Process:
1. **Synthetic Division Setup:**
- The polynomial is \( x^3 - 5x^2 + 3x + 1 \).
- The zero provided is \( x = 1 \).
2. **Perform Synthetic Division:**
- We'll set up synthetic division with \( 1 \) and the coefficients of the polynomial: \( [1, -5, 3, 1] \).
3. **Step-by-Step Synthetic Division:**
- Bring down the leading coefficient \( 1 \).
- Multiply \( 1 \) by \( 1 \) (zero given) and add to the next coefficient (\(-5\)): \( 1 \cdot 1 + (-5) = -4 \).
- Repeat this process until all coefficients are used. This will give us the quotient polynomial and the remainder.
- The resulting expression should help us write \( f(x) \) as a product of linear factors.
### Final Expression:
Once the division is performed correctly, the final expression will be in the form:
\[ f(x) = (x - 1)(\text{other factors}) \]
### Educational Context:
This problem helps students understand how to find factors of polynomials using given roots and zeroes. Understanding this method is crucial for simplifying higher-order polynomials and can be applied in further algebraic manipulations and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64f54e3f-bbb2-46db-a1a9-bb995302d054%2Fc3cad434-ab17-4400-971a-1e80976a4848%2F3ozjig.png&w=3840&q=75)
Transcribed Image Text:### Polynomial Factorization
**For the polynomial below, 1 is a zero.**
\[ f(x) = x^3 - 5x^2 + 3x + 1 \]
**Express \( f(x) \) as a product of linear factors.**
\[ f(x) = \]
In the image, you can also see an interactive element (probably from an educational platform) showing an icon with three options and the lowercase letter "i" along with typical icons for "ok," "reset," and "help." This likely indicates that the user is expected to input their answer or select different functionalities to assist in solving the problem.
### Explanation:
To solve this problem, follow these steps:
1. **Identify the given zero:** We are told that \(1\) is a zero of the polynomial.
2. **Polynomial Division:** Use synthetic or long division to divide the polynomial \( f(x) \) by \( x - 1 \).
3. **Factorization:** Express the quotient as a product of linear factors.
### Process:
1. **Synthetic Division Setup:**
- The polynomial is \( x^3 - 5x^2 + 3x + 1 \).
- The zero provided is \( x = 1 \).
2. **Perform Synthetic Division:**
- We'll set up synthetic division with \( 1 \) and the coefficients of the polynomial: \( [1, -5, 3, 1] \).
3. **Step-by-Step Synthetic Division:**
- Bring down the leading coefficient \( 1 \).
- Multiply \( 1 \) by \( 1 \) (zero given) and add to the next coefficient (\(-5\)): \( 1 \cdot 1 + (-5) = -4 \).
- Repeat this process until all coefficients are used. This will give us the quotient polynomial and the remainder.
- The resulting expression should help us write \( f(x) \) as a product of linear factors.
### Final Expression:
Once the division is performed correctly, the final expression will be in the form:
\[ f(x) = (x - 1)(\text{other factors}) \]
### Educational Context:
This problem helps students understand how to find factors of polynomials using given roots and zeroes. Understanding this method is crucial for simplifying higher-order polynomials and can be applied in further algebraic manipulations and
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