Which Polynomial Function Has A Leading Coefficient Of 1 And Roots \[$(7+i)\$\] And \[$(5-i)\$\] With Multiplicity 1?A. \[$f(x) = (x+7)(x-i)(x+5)(x+i)\$\]B. \[$f(x) = (x-7)(x-i)(x-5)(x+i)\$\]C. \[$f(x) =
Introduction
In mathematics, polynomial functions are a fundamental concept in algebra and are used to model various real-world phenomena. A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and the variables are raised to non-negative integer powers. In this article, we will explore the concept of polynomial functions with a leading coefficient of 1 and roots with multiplicity 1.
What are Roots in Polynomial Functions?
Roots of a polynomial function are the values of the variable that make the function equal to zero. In other words, if we have a polynomial function f(x), then the roots of f(x) are the values of x that satisfy the equation f(x) = 0. The multiplicity of a root is the number of times the root appears in the factorization of the polynomial function.
The Given Problem
The problem states that we have a polynomial function with a leading coefficient of 1 and roots (7+i) and (5-i) with multiplicity 1. We need to determine which of the given options is the correct polynomial function.
Option A: f(x) = (x+7)(x-i)(x+5)(x+i)
Let's analyze option A. The given roots are (7+i) and (5-i), which are complex conjugates. This means that the polynomial function must have real coefficients. The product of the factors (x+7) and (x+5) is (x^2 + 12x + 35), which is a quadratic expression with real coefficients. Similarly, the product of the factors (x-i) and (x+i) is (x^2 + 1), which is also a quadratic expression with real coefficients.
Option B: f(x) = (x-7)(x-i)(x-5)(x+i)
Now, let's analyze option B. The given roots are (7+i) and (5-i), which are complex conjugates. This means that the polynomial function must have real coefficients. The product of the factors (x-7) and (x-5) is (x^2 - 12x + 35), which is a quadratic expression with real coefficients. Similarly, the product of the factors (x-i) and (x+i) is (x^2 + 1), which is also a quadratic expression with real coefficients.
Option C: f(x) = (x-7)(x-i)(x+5)(x+i)
Now, let's analyze option C. The given roots are (7+i) and (5-i), which are complex conjugates. This means that the polynomial function must have real coefficients. The product of the factors (x-7) and (x+5) is (x^2 - 2x - 35), which is a quadratic expression with real coefficients. Similarly, the product of the factors (x-i) and (x+i) is (x^2 + 1), which is also a quadratic expression with real coefficients.
Conclusion
Based on the analysis of the given options, we can conclude that the correct polynomial function is option A: f(x) = (x+7)(x-i)(x+5)(x+i). This polynomial function has a leading coefficient of 1 and roots (7+i) and (5-i) with multiplicity 1.
Why is Option A the Correct Answer?
Option A is the correct answer because it satisfies the given conditions. The polynomial function has a leading coefficient of 1, and the roots (7+i) and (5-i) are complex conjugates, which means that the polynomial function must have real coefficients. The product of the factors (x+7) and (x+5) is (x^2 + 12x + 35), which is a quadratic expression with real coefficients. Similarly, the product of the factors (x-i) and (x+i) is (x^2 + 1), which is also a quadratic expression with real coefficients.
What are the Implications of this Result?
The result of this problem has implications for various fields, including mathematics, physics, and engineering. In mathematics, this result demonstrates the importance of complex conjugates in polynomial functions. In physics, this result has implications for the study of wave functions and quantum mechanics. In engineering, this result has implications for the design of filters and signal processing systems.
Final Thoughts
Q&A: Polynomial Functions and Roots
Q: What is a polynomial function?
A: A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and the variables are raised to non-negative integer powers.
Q: What are roots in polynomial functions?
A: Roots of a polynomial function are the values of the variable that make the function equal to zero. In other words, if we have a polynomial function f(x), then the roots of f(x) are the values of x that satisfy the equation f(x) = 0.
Q: What is the multiplicity of a root?
A: The multiplicity of a root is the number of times the root appears in the factorization of the polynomial function.
Q: What are complex conjugates?
A: Complex conjugates are pairs of complex numbers that have the same real part and opposite imaginary parts. For example, (3+4i) and (3-4i) are complex conjugates.
Q: Why are complex conjugates important in polynomial functions?
A: Complex conjugates are important in polynomial functions because they ensure that the polynomial function has real coefficients. When we multiply two complex conjugates, we get a real number.
Q: How do we find the roots of a polynomial function?
A: We can find the roots of a polynomial function by factoring the polynomial function into linear factors. Each linear factor corresponds to a root of the polynomial function.
Q: What is the difference between a root and a zero of a polynomial function?
A: A root of a polynomial function is a value of the variable that makes the function equal to zero. A zero of a polynomial function is a value of the variable that makes the function equal to zero, but it can also be a value that makes the function equal to a non-zero constant.
Q: Can a polynomial function have multiple roots?
A: Yes, a polynomial function can have multiple roots. In fact, a polynomial function can have multiple roots of the same multiplicity.
Q: How do we determine the multiplicity of a root?
A: We can determine the multiplicity of a root by factoring the polynomial function into linear factors. Each linear factor corresponds to a root of the polynomial function, and the multiplicity of the root is the number of times the linear factor appears in the factorization.
Q: What are some common applications of polynomial functions?
A: Polynomial functions have many applications in mathematics, physics, and engineering. Some common applications include:
- Modeling population growth and decay
- Analyzing electrical circuits
- Designing filters and signal processing systems
- Studying wave functions and quantum mechanics
Q: Can you provide some examples of polynomial functions?
A: Yes, here are some examples of polynomial functions:
- f(x) = x^2 + 3x + 2
- f(x) = x^3 - 2x^2 + x + 1
- f(x) = x^4 + 2x^3 - 3x^2 + x - 1
Q: How do we graph polynomial functions?
A: We can graph polynomial functions by plotting the x-intercepts and the y-intercepts of the function. We can also use technology, such as graphing calculators or computer software, to graph polynomial functions.
Q: What are some common mistakes to avoid when working with polynomial functions?
A: Some common mistakes to avoid when working with polynomial functions include:
- Not factoring the polynomial function correctly
- Not identifying the roots of the polynomial function correctly
- Not determining the multiplicity of the roots correctly
- Not graphing the polynomial function correctly
Conclusion
In conclusion, polynomial functions and roots are fundamental concepts in mathematics that have many applications in physics and engineering. By understanding the properties and behavior of polynomial functions, we can model and analyze complex systems and phenomena.