A polynomial f (x) and two of its zeros are given. f(x)=2x° +11x* +22x' – 62x² – 120x+72; 1 -3-3i and are zeros 2
A polynomial f (x) and two of its zeros are given. f(x)=2x° +11x* +22x' – 62x² – 120x+72; 1 -3-3i and are zeros 2
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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A polynomial \( f(x) \) and two of its zeros are given.
\[ f(x) = 2x^5 + 11x^4 + 22x^3 - 62x^2 - 120x + 72; \]
\[ -3 - 3i \quad \text{and} \quad \frac{1}{2} \quad \text{are zeros} \]
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This excerpt presents a polynomial function \( f(x) \) along with two of its zeros. The polynomial is of the fifth degree, given by:
\[ f(x) = 2x^5 + 11x^4 + 22x^3 - 62x^2 - 120x + 72; \]
Additionally, the zeros of the polynomial provided are:
\[ -3 - 3i \quad \text{and} \quad \frac{1}{2} \]
In this context, a zero of a polynomial is a value of \( x \) for which \( f(x) = 0 \).
**Note for Students**: Remember that if a polynomial has a complex zero \( a + bi \), the conjugate \( a - bi \) is also a zero if the coefficients of the polynomial are real.
Understanding these concepts is crucial in solving polynomial equations and finding all the roots (zeros) of the given polynomial function.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F89b0c760-1322-44b8-ad46-a0dd06562c0d%2F3af5ec67-36e1-462a-bbd2-bd84c1d84bae%2Ffumuv9g_processed.png&w=3840&q=75)
Transcribed Image Text:---
A polynomial \( f(x) \) and two of its zeros are given.
\[ f(x) = 2x^5 + 11x^4 + 22x^3 - 62x^2 - 120x + 72; \]
\[ -3 - 3i \quad \text{and} \quad \frac{1}{2} \quad \text{are zeros} \]
---
This excerpt presents a polynomial function \( f(x) \) along with two of its zeros. The polynomial is of the fifth degree, given by:
\[ f(x) = 2x^5 + 11x^4 + 22x^3 - 62x^2 - 120x + 72; \]
Additionally, the zeros of the polynomial provided are:
\[ -3 - 3i \quad \text{and} \quad \frac{1}{2} \]
In this context, a zero of a polynomial is a value of \( x \) for which \( f(x) = 0 \).
**Note for Students**: Remember that if a polynomial has a complex zero \( a + bi \), the conjugate \( a - bi \) is also a zero if the coefficients of the polynomial are real.
Understanding these concepts is crucial in solving polynomial equations and finding all the roots (zeros) of the given polynomial function.
---
![### Factoring \( f(x) \) as a Product of Linear Factors
#### Problem Statement
(b) Factor \( f(x) \) as a product of linear factors.
\[ f(x) = \]
#### Explanation
To solve this problem, you need to express the function \( f(x) \) as a product of linear factors. A linear factor is generally in the form \((x - a)\), where \(a\) is a root of the equation.
#### Input Box and Tool Menu
In the provided image, there appears to be an empty input box next to \( f(x) = \). This is where you will input the factored form of the function.
Additionally, there is a tool menu with various mathematical symbols:
1. **i**: Likely to represent the imaginary unit.
2. **Fraction Symbol**: Indicates the option to input fractions.
3. **Power Symbol**: Allows you to input exponents.
4. **Square Root Symbol**: Permits entry of square roots.
#### Graphs or Diagrams
There are no graphs or diagrams associated with this problem statement.
#### Steps to Factor \( f(x) \)
1. Identify the roots of the polynomial \( f(x) \). This could be done through techniques such as solving the equation \( f(x) = 0 \), synthetic division, or factoring by grouping.
2. Once the roots \( r_1, r_2, \dots, r_n \) are identified, express \( f(x) \) as:
\[ f(x) = a(x - r_1)(x - r_2) \cdots (x - r_n) \]
Where \( a \) is the leading coefficient of the polynomial.
#### Example
For instance, if the polynomial is \( f(x) = x^2 - 5x + 6 \), you can find that the roots are \( x = 2 \) and \( x = 3 \). Thus, the factorized form is:
\[ f(x) = (x - 2)(x - 3) \]
Remember, the exact factorization depends on the given function \( f(x) \). The provided tools can help ensure correct formatting of mathematical expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F89b0c760-1322-44b8-ad46-a0dd06562c0d%2F3af5ec67-36e1-462a-bbd2-bd84c1d84bae%2Fugcus9c_processed.png&w=3840&q=75)
Transcribed Image Text:### Factoring \( f(x) \) as a Product of Linear Factors
#### Problem Statement
(b) Factor \( f(x) \) as a product of linear factors.
\[ f(x) = \]
#### Explanation
To solve this problem, you need to express the function \( f(x) \) as a product of linear factors. A linear factor is generally in the form \((x - a)\), where \(a\) is a root of the equation.
#### Input Box and Tool Menu
In the provided image, there appears to be an empty input box next to \( f(x) = \). This is where you will input the factored form of the function.
Additionally, there is a tool menu with various mathematical symbols:
1. **i**: Likely to represent the imaginary unit.
2. **Fraction Symbol**: Indicates the option to input fractions.
3. **Power Symbol**: Allows you to input exponents.
4. **Square Root Symbol**: Permits entry of square roots.
#### Graphs or Diagrams
There are no graphs or diagrams associated with this problem statement.
#### Steps to Factor \( f(x) \)
1. Identify the roots of the polynomial \( f(x) \). This could be done through techniques such as solving the equation \( f(x) = 0 \), synthetic division, or factoring by grouping.
2. Once the roots \( r_1, r_2, \dots, r_n \) are identified, express \( f(x) \) as:
\[ f(x) = a(x - r_1)(x - r_2) \cdots (x - r_n) \]
Where \( a \) is the leading coefficient of the polynomial.
#### Example
For instance, if the polynomial is \( f(x) = x^2 - 5x + 6 \), you can find that the roots are \( x = 2 \) and \( x = 3 \). Thus, the factorized form is:
\[ f(x) = (x - 2)(x - 3) \]
Remember, the exact factorization depends on the given function \( f(x) \). The provided tools can help ensure correct formatting of mathematical expressions.
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