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 Division and Factoring Problems
#### 1. Polynomial Division
**Problem:**
Divide \(x^4 + 4x^3 + 16x - 35\) by \(x + 5\).
**Steps to Solve:**
1. Perform polynomial long division or synthetic division.
2. Identify the quotient and the remainder.
**Solution Process:**
```
x^3 - x^2 + 5x - 18
___________________
x + 5 | x^4 + 4x^3 + 0x^2 + 16x - 35
- (x^4 + 5x^3)
___________________
-x^3 + 0x^2
- (-x^3 - 5x^2)
___________________
5x^2 + 16x
- ( 5x^2 + 25x)
___________________
- 9x - 35
- ( -9x - 45)
___________________
10
```
**Quotient:**
\[ x^3 - x^2 + 5x - 18 \]
**Remainder:**
\[ 10 \]
#### 2. Polynomial Factoring
**Problem:**
A polynomial and one of its factors is given. Factor the polynomial **completely** given that one of its factors is \(x + 4\). Your final answer should be in factored form.
\[ f(x) = x^7 + 4x^6 + 7x^4 + 28x^3 - 8x - 32 \]
**Solution Process:**
1. Given factor: \(x + 4\).
2. Use polynomial division or synthetic division to divide \(f(x)\) by \(x + 4\).
3. Further factor the quotient if necessary.
**Factored form:**
\[ f(x) = (x + 4)(\text{Quotients}) \]
(Note: Since the full solution involves polynomial division, it's recommended to solve this using algebraic long division or synthetic division methods.)
#### 3. Finding Real Zeros
**Problem:**
Find **ALL** additional real zeros of the given polynomial equation given that \(x = 6\) is a zero of the polynomial. Your final](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3fd9fba-5a8f-483b-ab28-e1e51fab3fbe%2F85c7e487-3b40-4c65-9004-88d445d0b239%2Fezd9iyd_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Polynomial Division and Factoring Problems
#### 1. Polynomial Division
**Problem:**
Divide \(x^4 + 4x^3 + 16x - 35\) by \(x + 5\).
**Steps to Solve:**
1. Perform polynomial long division or synthetic division.
2. Identify the quotient and the remainder.
**Solution Process:**
```
x^3 - x^2 + 5x - 18
___________________
x + 5 | x^4 + 4x^3 + 0x^2 + 16x - 35
- (x^4 + 5x^3)
___________________
-x^3 + 0x^2
- (-x^3 - 5x^2)
___________________
5x^2 + 16x
- ( 5x^2 + 25x)
___________________
- 9x - 35
- ( -9x - 45)
___________________
10
```
**Quotient:**
\[ x^3 - x^2 + 5x - 18 \]
**Remainder:**
\[ 10 \]
#### 2. Polynomial Factoring
**Problem:**
A polynomial and one of its factors is given. Factor the polynomial **completely** given that one of its factors is \(x + 4\). Your final answer should be in factored form.
\[ f(x) = x^7 + 4x^6 + 7x^4 + 28x^3 - 8x - 32 \]
**Solution Process:**
1. Given factor: \(x + 4\).
2. Use polynomial division or synthetic division to divide \(f(x)\) by \(x + 4\).
3. Further factor the quotient if necessary.
**Factored form:**
\[ f(x) = (x + 4)(\text{Quotients}) \]
(Note: Since the full solution involves polynomial division, it's recommended to solve this using algebraic long division or synthetic division methods.)
#### 3. Finding Real Zeros
**Problem:**
Find **ALL** additional real zeros of the given polynomial equation given that \(x = 6\) is a zero of the polynomial. Your final
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