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
![### Problem 4
**Task:** Use synthetic division to factor the polynomial \( x^3 + 3x^2 - 10x - 24 \) completely if 3 is a zero.
**Instructions:** Apply synthetic division with the provided zero to factor the polynomial. Show your steps clearly.
**Answer:** [Place the answer here, showing each step of the synthetic division process and the resulting factors.]
### Explanation of Method:
1. **Set Up Synthetic Division:**
- Use 3 (the zero) to set up the division.
- Write the coefficients of the polynomial: 1 (for \(x^3\)), 3 (for \(x^2\)), -10 (for \(x\)), and -24 (the constant).
2. **Perform Synthetic Division:**
- Bring down the leading coefficient.
- Multiply the zero by this coefficient and add to the next coefficient.
- Continue this process across all coefficients.
3. **Interpret the Result:**
- The final line in synthetic division gives the coefficients of the quotient polynomial.
- If any remainder exists, it will be the final value.
4. **Factor Completely:**
- Use the quotient to factor further if necessary.
This process helps simplify and completely factor the given polynomial, using \( x - 3 \) as one of the factors due to the given zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa610010-7ad2-4d49-b949-5d449ad446df%2F3023c78c-8b79-4678-aae6-2db08ef9716b%2Fczsr9ep_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 4
**Task:** Use synthetic division to factor the polynomial \( x^3 + 3x^2 - 10x - 24 \) completely if 3 is a zero.
**Instructions:** Apply synthetic division with the provided zero to factor the polynomial. Show your steps clearly.
**Answer:** [Place the answer here, showing each step of the synthetic division process and the resulting factors.]
### Explanation of Method:
1. **Set Up Synthetic Division:**
- Use 3 (the zero) to set up the division.
- Write the coefficients of the polynomial: 1 (for \(x^3\)), 3 (for \(x^2\)), -10 (for \(x\)), and -24 (the constant).
2. **Perform Synthetic Division:**
- Bring down the leading coefficient.
- Multiply the zero by this coefficient and add to the next coefficient.
- Continue this process across all coefficients.
3. **Interpret the Result:**
- The final line in synthetic division gives the coefficients of the quotient polynomial.
- If any remainder exists, it will be the final value.
4. **Factor Completely:**
- Use the quotient to factor further if necessary.
This process helps simplify and completely factor the given polynomial, using \( x - 3 \) as one of the factors due to the given zero.
![**Transcription for Educational Website**
---
**Example Problem: Finding Zeros of a Polynomial**
**Question:**
Find all the zeros of the polynomial:
\[ x^4 - 6x^3 + 14x^2 - 54x + 45 = 0. \]
**Solution Process:**
1. **Factoring and Solving:**
The given polynomial can be analyzed through methods like synthetic division or factoring to identify its zeros. One way to identify potential zeros is to test for integer solutions and use polynomial division to simplify.
2. **Identified Zeros:**
By calculating, we find:
\[ x = 1, \, x = 3, \, x = -3. \]
These are the zeros where the polynomial equals zero, representing the x-values where the polynomial intersects the x-axis.
3. **Steps and Justification:**
The polynomial is first checked for possible factorization or root patterns. After evaluation, the identified values satisfy:
\[ 1 - 6 + 14 - 54 + 45 = 0. \]
**Further Practice:**
**Question:**
Find a fourth-degree polynomial with zeros at 3, -3, and additional complex roots \( \pm i \).
**Answer:**
A polynomial with these zeros can be constructed as:
\[ (x - 3)(x + 3)(x - i)(x + i). \]
This equals:
\[ (x^2 - 9)(x^2 + 1), \]
resulting in:
\[ x^4 - 8x^2 + 9. \]
---
In this example, polynomial factorization is used to find zeros, offering an insight into techniques used for higher-degree polynomials. Understanding these methods is crucial for solving algebraic equations and analyzing their graphs.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa610010-7ad2-4d49-b949-5d449ad446df%2F3023c78c-8b79-4678-aae6-2db08ef9716b%2Fhrp4u9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website**
---
**Example Problem: Finding Zeros of a Polynomial**
**Question:**
Find all the zeros of the polynomial:
\[ x^4 - 6x^3 + 14x^2 - 54x + 45 = 0. \]
**Solution Process:**
1. **Factoring and Solving:**
The given polynomial can be analyzed through methods like synthetic division or factoring to identify its zeros. One way to identify potential zeros is to test for integer solutions and use polynomial division to simplify.
2. **Identified Zeros:**
By calculating, we find:
\[ x = 1, \, x = 3, \, x = -3. \]
These are the zeros where the polynomial equals zero, representing the x-values where the polynomial intersects the x-axis.
3. **Steps and Justification:**
The polynomial is first checked for possible factorization or root patterns. After evaluation, the identified values satisfy:
\[ 1 - 6 + 14 - 54 + 45 = 0. \]
**Further Practice:**
**Question:**
Find a fourth-degree polynomial with zeros at 3, -3, and additional complex roots \( \pm i \).
**Answer:**
A polynomial with these zeros can be constructed as:
\[ (x - 3)(x + 3)(x - i)(x + i). \]
This equals:
\[ (x^2 - 9)(x^2 + 1), \]
resulting in:
\[ x^4 - 8x^2 + 9. \]
---
In this example, polynomial factorization is used to find zeros, offering an insight into techniques used for higher-degree polynomials. Understanding these methods is crucial for solving algebraic equations and analyzing their graphs.
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