For the polynomial below, -1 is a zero of multiplicity two. 4 f(x)=x"+4x+10x+12x+5 Express f (x) as a product of linear factors. %3D
For the polynomial below, -1 is a zero of multiplicity two. 4 f(x)=x"+4x+10x+12x+5 Express f (x) as a product of linear factors. %3D
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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Question
![### Polynomial Zeroes and Factorization
For the polynomial below, \( -1 \) is a zero of **multiplicity two**.
\[ f(x) = x^4 + 4x^3 + 10x^2 + 12x + 5 \]
Express \( f(x) \) as a **product of linear factors**.
\[ f(x) = \]
### Explanation
The problem presents a polynomial function \( f(x) \) and states that \( -1 \) is a zero of multiplicity two. This means that \( (x + 1)^2 \) is a factor of the polynomial. The goal is to express the polynomial as a product of linear factors by identifying other zeroes and factoring completely.
For educational purposes, the steps for completing this problem would typically involve:
1. **Using Synthetic Division:** Perform synthetic division of the polynomial by \( x + 1 \) twice, as \(-1\) is a zero of multiplicity two.
2. **Finding Remaining Zeroes:** Once the polynomial is divided, solve the reduced polynomial to find remaining zeroes.
3. **Factoring Completely:** Express the polynomial as a product of factors based on the zeroes found from the division and solving.
### Important Concepts
- **Zero of Multiplicity:** A zero that occurs more than once in the factoring of a polynomial.
- **Linear Factor:** A factor of the form \( (x - c) \) where \( c \) is a zero of the polynomial.
- **Synthetic Division:** A shortcut method of polynomial division, particularly useful when dividing by linear factors.
This understanding is critical for students learning about polynomial functions, zeroes, and factorization.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29ce3a4c-cb07-4c4c-bb91-a57b2d1f76d3%2F7157d5dd-1acc-492e-acfc-66ff81959adf%2F14zb0de_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Polynomial Zeroes and Factorization
For the polynomial below, \( -1 \) is a zero of **multiplicity two**.
\[ f(x) = x^4 + 4x^3 + 10x^2 + 12x + 5 \]
Express \( f(x) \) as a **product of linear factors**.
\[ f(x) = \]
### Explanation
The problem presents a polynomial function \( f(x) \) and states that \( -1 \) is a zero of multiplicity two. This means that \( (x + 1)^2 \) is a factor of the polynomial. The goal is to express the polynomial as a product of linear factors by identifying other zeroes and factoring completely.
For educational purposes, the steps for completing this problem would typically involve:
1. **Using Synthetic Division:** Perform synthetic division of the polynomial by \( x + 1 \) twice, as \(-1\) is a zero of multiplicity two.
2. **Finding Remaining Zeroes:** Once the polynomial is divided, solve the reduced polynomial to find remaining zeroes.
3. **Factoring Completely:** Express the polynomial as a product of factors based on the zeroes found from the division and solving.
### Important Concepts
- **Zero of Multiplicity:** A zero that occurs more than once in the factoring of a polynomial.
- **Linear Factor:** A factor of the form \( (x - c) \) where \( c \) is a zero of the polynomial.
- **Synthetic Division:** A shortcut method of polynomial division, particularly useful when dividing by linear factors.
This understanding is critical for students learning about polynomial functions, zeroes, and factorization.
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