9) Given that 1 + 5i is a zero of f(x)=x² - 3x² +28x-26 (a) find the remaining zeros 30erit worle of msto (b) factor f(x)as a product of linear factors.
9) Given that 1 + 5i is a zero of f(x)=x² - 3x² +28x-26 (a) find the remaining zeros 30erit worle of msto (b) factor f(x)as a product of linear factors.
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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![### Polynomial Zeros and Factoring
#### Question 9
Given that \(1 + 5i\) is a zero of \( f(x) = x^3 - 3x^2 + 28x - 26 \):
(a) Find the remaining zeros.
(b) Factor \( f(x) \) as a product of linear factors.
#### Solution Guide
**Step-by-Step Instructions:**
1. **Understanding the Polynomial:**
- The polynomial is given as:
\[
f(x) = x^3 - 3x^2 + 28x - 26
\]
2. **Given Zero:**
- One of the zeros is given as \(1 + 5i\).
3. **Conjugate Pairs:**
- For polynomials with real coefficients, if \(1 + 5i\) is a zero, its conjugate \(1 - 5i\) must also be a zero.
4. **Finding the Third Zero:**
- Since the polynomial is cubic (degree 3), it must have three zeros.
- Let the third zero be \(r\).
5. **Product of Zeros:**
- Since we know two zeros (\(1 + 5i\) and \(1 - 5i\)), we use polynomial division or synthetic division to find the remaining zero.
6. **Factoring Process:**
- After finding all zeros, express the polynomial \( f(x) \) as the product of its linear factors.
### Detailed Explanation of Concepts:
1. **Zeros of a Polynomial:**
- A zero of a polynomial is a value of \(x\) for which the polynomial equals zero.
- For complex zeros, they often come in conjugate pairs when coefficients of the polynomial are real.
2. **Conjugate Pairs:**
- If a polynomial has real coefficients, and \(a + bi\) is a zero, then \(a - bi\) must also be a zero.
3. **Factoring Polynomials:**
- Factoring a polynomial means expressing it as a product of lower-degree polynomials.
- For a cubic polynomial \(ax^3 + bx^2 + cx + d\), factoring will break it down into linear factors \((x - p)(x - q)(x - r)\) where \(p, q, r\) are the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F265296a1-da32-4f4b-bf09-d4ccbee1822d%2F56dac6d7-02ae-40e1-8952-9b424fceb2aa%2Fp0trhkf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Polynomial Zeros and Factoring
#### Question 9
Given that \(1 + 5i\) is a zero of \( f(x) = x^3 - 3x^2 + 28x - 26 \):
(a) Find the remaining zeros.
(b) Factor \( f(x) \) as a product of linear factors.
#### Solution Guide
**Step-by-Step Instructions:**
1. **Understanding the Polynomial:**
- The polynomial is given as:
\[
f(x) = x^3 - 3x^2 + 28x - 26
\]
2. **Given Zero:**
- One of the zeros is given as \(1 + 5i\).
3. **Conjugate Pairs:**
- For polynomials with real coefficients, if \(1 + 5i\) is a zero, its conjugate \(1 - 5i\) must also be a zero.
4. **Finding the Third Zero:**
- Since the polynomial is cubic (degree 3), it must have three zeros.
- Let the third zero be \(r\).
5. **Product of Zeros:**
- Since we know two zeros (\(1 + 5i\) and \(1 - 5i\)), we use polynomial division or synthetic division to find the remaining zero.
6. **Factoring Process:**
- After finding all zeros, express the polynomial \( f(x) \) as the product of its linear factors.
### Detailed Explanation of Concepts:
1. **Zeros of a Polynomial:**
- A zero of a polynomial is a value of \(x\) for which the polynomial equals zero.
- For complex zeros, they often come in conjugate pairs when coefficients of the polynomial are real.
2. **Conjugate Pairs:**
- If a polynomial has real coefficients, and \(a + bi\) is a zero, then \(a - bi\) must also be a zero.
3. **Factoring Polynomials:**
- Factoring a polynomial means expressing it as a product of lower-degree polynomials.
- For a cubic polynomial \(ax^3 + bx^2 + cx + d\), factoring will break it down into linear factors \((x - p)(x - q)(x - r)\) where \(p, q, r\) are the
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