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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please help with this practice questions thank you! not graded!
![Certainly! Here is the transcribed content suitable for an educational website:
---
### Question 7
**Explain the possible ways to build a function of degree \( n \).**
---
To address this question:
1. **Polynomial Definition**:
A function of degree \( n \) is typically a polynomial function, which can be expressed in the general form:
\[
f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0
\]
where \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are constants, and \( a_n \neq 0 \).
2. **Building the Function**:
- **Identify the Degree**: The degree \( n \) is the highest power of \( x \) with a non-zero coefficient.
- **Choose Coefficients**: Select constant coefficients \( a_n, a_{n-1}, \ldots, a_0 \) as desired. The leading coefficient \( a_n \) should not be zero.
- **Select Roots/Factors**: A polynomial can also be constructed by specifying its roots and using the factor theorem. For example, if the roots are \( r_1, r_2, \ldots, r_n \), the polynomial can be expressed as:
\[
f(x) = a_n(x - r_1)(x - r_2)\cdots(x - r_n)
\]
- **Sum of Functions**: Create a function of degree \( n \) by summing or combining other functions. For instance, add a quadratic and linear function to create a cubic function.
3. **Consider Special Functions**:
- **Monomials**: Simple functions such as \( f(x) = x^n \) are monomials of degree \( n \).
- **Binomials and Trinomials**: Build using fewer terms like \( f(x) = 3x^n + 2 \).
4. **Graphical Representation**:
- Identifying critical features like intercepts, turning points, and behavior as \( x \to \pm \infty \), can guide in modifying the function form for specific properties.
By understanding these methods, you can effectively build and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9aebe04-9188-4807-af82-86075f7359a2%2F6d9e310a-5da8-44f2-a01b-2c496a34e8cf%2F5aw5nrw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here is the transcribed content suitable for an educational website:
---
### Question 7
**Explain the possible ways to build a function of degree \( n \).**
---
To address this question:
1. **Polynomial Definition**:
A function of degree \( n \) is typically a polynomial function, which can be expressed in the general form:
\[
f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0
\]
where \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are constants, and \( a_n \neq 0 \).
2. **Building the Function**:
- **Identify the Degree**: The degree \( n \) is the highest power of \( x \) with a non-zero coefficient.
- **Choose Coefficients**: Select constant coefficients \( a_n, a_{n-1}, \ldots, a_0 \) as desired. The leading coefficient \( a_n \) should not be zero.
- **Select Roots/Factors**: A polynomial can also be constructed by specifying its roots and using the factor theorem. For example, if the roots are \( r_1, r_2, \ldots, r_n \), the polynomial can be expressed as:
\[
f(x) = a_n(x - r_1)(x - r_2)\cdots(x - r_n)
\]
- **Sum of Functions**: Create a function of degree \( n \) by summing or combining other functions. For instance, add a quadratic and linear function to create a cubic function.
3. **Consider Special Functions**:
- **Monomials**: Simple functions such as \( f(x) = x^n \) are monomials of degree \( n \).
- **Binomials and Trinomials**: Build using fewer terms like \( f(x) = 3x^n + 2 \).
4. **Graphical Representation**:
- Identifying critical features like intercepts, turning points, and behavior as \( x \to \pm \infty \), can guide in modifying the function form for specific properties.
By understanding these methods, you can effectively build and
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