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.
Introduction
In algebra, polynomial functions are a fundamental concept that plays a crucial role in various mathematical operations. A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. In this article, we will focus on polynomial functions with complex roots, specifically those with a leading coefficient of 1 and roots {(7+i)$}$ and {(5-i)$}$ with multiplicity 1.
Understanding Complex Roots
Complex roots are roots of a polynomial equation that are not real numbers. They are denoted by the letter "i," which represents the imaginary unit, where i^2 = -1. Complex roots always come in conjugate pairs, meaning that if a polynomial has a complex root, its conjugate is also a root. In this case, the roots {(7+i)$}$ and {(5-i)$}$ are conjugates of each other.
The Leading Coefficient
The leading coefficient of a polynomial function is the coefficient of the highest degree term. In this case, the leading coefficient is 1, which means that the polynomial function has no constant term. This implies that the polynomial function has a degree of at least 2, as the highest degree term is x^2.
Multiplicity of Roots
The multiplicity of a root is the number of times the root appears in the factorization of the polynomial function. In this case, the roots {(7+i)$}$ and {(5-i)$}$ have a multiplicity of 1, meaning that they appear only once in the factorization of the polynomial function.
Analyzing the Options
Now that we have a clear understanding of the properties of the polynomial function, let's analyze the options provided:
Option A: {f(x)=(x+7)(x-i)(x+5)(x+i)$}$
This option represents a polynomial function with roots {(7+i)$}$ and {(5-i)$}$ with multiplicity 1. However, the leading coefficient is not 1, as the constant term is not zero. Therefore, this option is not a valid solution.
Option B: {f(x)=(x-7)(x-i)(x-5)(x+i)$}$
This option represents a polynomial function with roots {(7+i)$}$ and {(5-i)$}$ with multiplicity 1. However, the leading coefficient is not 1, as the constant term is not zero. Therefore, this option is not a valid solution.
Option C: {f(x)=(x-(7+i))(x-(5-i))$}$
This option represents a polynomial function with roots {(7+i)$}$ and {(5-i)$}$ with multiplicity 1. However, the leading coefficient is not 1, as the constant term is not zero. Therefore, this option is not a valid solution.
The Correct Solution
After analyzing the options, we can see that none of them represent a polynomial function with a leading coefficient of 1 and roots {(7+i)$}$ and {(5-i)$}$ with multiplicity 1. However, we can find the correct solution by multiplying the factors {(x-(7+i))$}$ and {(x-(5-i))$}$ to obtain the polynomial function:
{f(x)=(x-(7+i))(x-(5-i))$}$
{f(x)=(x-7-i)(x-5+i)$}$
{f(x)=x^2-12x+50-2i$}$
{f(x)=x^2-12x+50-2i$}$
However, this is not the correct solution. We need to multiply the factors by their conjugates to obtain the correct solution:
{f(x)=(x-(7+i))(x-(5-i))(x-(7-i))(x-(5+i))$}$
{f(x)=(x-7-i)(x-5+i)(x-7+i)(x-5-i)$}$
{f(x)=(x2-12x+50-2i)(x2-12x+50+2i)$}$
{f(x)=x4-24x3+194x^2-600x+625$}$
This is the correct solution, which represents a polynomial function with a leading coefficient of 1 and roots {(7+i)$}$ and {(5-i)$}$ with multiplicity 1.
Conclusion
In conclusion, we have analyzed the properties of polynomial functions with complex roots and found the correct solution to the problem. The correct solution represents a polynomial function with a leading coefficient of 1 and roots {(7+i)$}$ and {(5-i)$}$ with multiplicity 1. This solution is essential in various mathematical operations, including algebraic manipulations and numerical computations.
References
- [1] "Polynomial Functions" by Math Open Reference
- [2] "Complex Roots" by Wolfram MathWorld
- [3] "Leading Coefficient" by Khan Academy
Further Reading
For further reading on polynomial functions and complex roots, we recommend the following resources:
- "Polynomial Functions" by MIT OpenCourseWare
- "Complex Analysis" by Stanford University
- "Algebra" by University of California, Berkeley
Note: The references and further reading resources are provided for informational purposes only and are not an endorsement of any particular resource or institution.
Introduction
In our previous article, we discussed polynomial functions with complex roots, specifically those with a leading coefficient of 1 and roots {(7+i)$}$ and {(5-i)$}$ with multiplicity 1. In this article, we will provide a Q&A section to address common questions and concerns related to polynomial functions with complex roots.
Q: What are complex roots?
A: Complex roots are roots of a polynomial equation that are not real numbers. They are denoted by the letter "i," which represents the imaginary unit, where i^2 = -1. Complex roots always come in conjugate pairs, meaning that if a polynomial has a complex root, its conjugate is also a root.
Q: What is the difference between a real root and a complex root?
A: A real root is a root of a polynomial equation that is a real number, whereas a complex root is a root that is not a real number. Complex roots are denoted by the letter "i," which represents the imaginary unit.
Q: How do I find the conjugate of a complex root?
A: To find the conjugate of a complex root, you need to change the sign of the imaginary part. For example, the conjugate of {(7+i)$}$ is {(7-i)$}$.
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. In this case, the roots {(7+i)$}$ and {(5-i)$}$ have a multiplicity of 1, meaning that they appear only once in the factorization of the polynomial function.
Q: How do I find the polynomial function with a leading coefficient of 1 and roots {(7+i)$}$ and {(5-i)$}$ with multiplicity 1?
A: To find the polynomial function, you need to multiply the factors {(x-(7+i))$}$ and {(x-(5-i))$}$ to obtain the polynomial function:
{f(x)=(x-(7+i))(x-(5-i))$}$
{f(x)=(x-7-i)(x-5+i)$}$
{f(x)=x^2-12x+50-2i$}$
However, this is not the correct solution. You need to multiply the factors by their conjugates to obtain the correct solution:
{f(x)=(x-(7+i))(x-(5-i))(x-(7-i))(x-(5+i))$}$
{f(x)=(x-7-i)(x-5+i)(x-7+i)(x-5-i)$}$
{f(x)=(x2-12x+50-2i)(x2-12x+50+2i)$}$
{f(x)=x4-24x3+194x^2-600x+625$}$
Q: What is the degree of a polynomial function with complex roots?
A: The degree of a polynomial function with complex roots is the highest power of the variable in the polynomial function. In this case, the degree of the polynomial function is 4.
Q: Can a polynomial function have a leading coefficient of 1 and roots {(7+i)$}$ and {(5-i)$}$ with multiplicity 1?
A: Yes, a polynomial function can have a leading coefficient of 1 and roots {(7+i)$}$ and {(5-i)$}$ with multiplicity 1. The correct solution is:
{f(x)=(x2-12x+50-2i)(x2-12x+50+2i)$}$
{f(x)=x4-24x3+194x^2-600x+625$}$
Conclusion
In conclusion, we have provided a Q&A section to address common questions and concerns related to polynomial functions with complex roots. We hope that this article has been helpful in understanding the properties of polynomial functions with complex roots and how to find the correct solution.
References
- [1] "Polynomial Functions" by Math Open Reference
- [2] "Complex Roots" by Wolfram MathWorld
- [3] "Leading Coefficient" by Khan Academy
Further Reading
For further reading on polynomial functions and complex roots, we recommend the following resources:
- "Polynomial Functions" by MIT OpenCourseWare
- "Complex Analysis" by Stanford University
- "Algebra" by University of California, Berkeley
Note: The references and further reading resources are provided for informational purposes only and are not an endorsement of any particular resource or institution.