Which Polynomial Function Has A Leading Coefficient Of 1 And Roots $2i$ And \$3i$[/tex\] With Multiplicity 1?A. $f(x) = (x-2)(x-3i$\]B. $f(x) = (x+2)(x+3$\]C. $f(x) = (x-2)(x-3)(x-2)(x-3$\]D. $f(x) =
Introduction
In algebra, a polynomial function is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The roots of a polynomial function are the values of the variable that make the function equal to zero. In this article, we will explore which polynomial function has a leading coefficient of 1 and roots 2i and 3i with multiplicity 1.
Understanding the Problem
To solve this problem, we need to understand the concept of roots and multiplicity. The roots of a polynomial function are the values of the variable that make the function equal to zero. The multiplicity of a root is the number of times that root appears in the factorization of the polynomial function.
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 highest degree term is x.
The Roots
The roots of the polynomial function are 2i and 3i. Since the multiplicity of each root is 1, we need to find a polynomial function that has these roots with multiplicity 1.
Factoring the Polynomial Function
To find the polynomial function, we can start by factoring the polynomial function using the roots. Since the roots are 2i and 3i, we can write the polynomial function as:
f(x) = (x - 2i)(x - 3i)
Multiplying the Factors
To find the polynomial function, we need to multiply the factors. Using the distributive property, we can multiply the factors as follows:
f(x) = (x - 2i)(x - 3i) = x^2 - 3ix - 2ix + 6i^2 = x^2 - 5ix + 6i^2
Simplifying the Expression
Since i^2 = -1, we can simplify the expression as follows:
f(x) = x^2 - 5ix - 6 = x^2 - 5ix + 6
Comparing the Options
Now that we have found the polynomial function, we can compare it to the options. The polynomial function we found is:
f(x) = x^2 - 5ix + 6
Comparing this to the options, we can see that option C is the correct answer.
Conclusion
In conclusion, the polynomial function that has a leading coefficient of 1 and roots 2i and 3i with multiplicity 1 is:
f(x) = (x - 2)(x - 3)(x - 2)(x - 3)
This polynomial function can be simplified to:
f(x) = x^2 - 5ix + 6
Therefore, the correct answer is option C.
Final Answer
The final answer is option C.
Additional Information
In this article, we have explored the concept of roots and multiplicity of a polynomial function. We have also seen how to factor a polynomial function using the roots and how to multiply the factors to find the polynomial function. This knowledge can be applied to a wide range of problems in algebra and other areas of mathematics.
References
- [1] "Algebra" by Michael Artin
- [2] "Polynomial Functions" by Wolfram MathWorld
Related Topics
- Roots of a polynomial function
- Multiplicity of a root
- Factoring a polynomial function
- Multiplying factors to find a polynomial function
Glossary
- Leading coefficient: The coefficient of the highest degree term in a polynomial function.
- Roots: The values of the variable that make a polynomial function equal to zero.
- Multiplicity: The number of times that a root appears in the factorization of a polynomial function.
- Factoring: The process of expressing a polynomial function as a product of simpler polynomial functions.
- Multiplying factors: The process of multiplying the factors of a polynomial function to find the polynomial function.
Introduction
In our previous article, we explored the concept of polynomial functions with a leading coefficient of 1 and roots 2i and 3i with multiplicity 1. We found that the polynomial function that satisfies these conditions is:
f(x) = (x - 2)(x - 3)(x - 2)(x - 3)
This polynomial function can be simplified to:
f(x) = x^2 - 5ix + 6
In this article, we will answer some frequently asked questions about polynomial functions with leading coefficient of 1 and roots 2i and 3i with multiplicity 1.
Q: What is the leading coefficient of a polynomial function?
A: The leading coefficient of a polynomial function is the coefficient of the highest degree term. In this case, the leading coefficient is 1.
Q: What are the roots of the polynomial function?
A: The roots of the polynomial function are 2i and 3i.
Q: What is the multiplicity of each root?
A: The multiplicity of each root is 1.
Q: How do I factor a polynomial function using the roots?
A: To factor a polynomial function using the roots, you can start by writing the polynomial function as a product of simpler polynomial functions. In this case, we can write the polynomial function as:
f(x) = (x - 2i)(x - 3i)
Q: How do I multiply the factors to find the polynomial function?
A: To multiply the factors, you can use the distributive property. In this case, we can multiply the factors as follows:
f(x) = (x - 2i)(x - 3i) = x^2 - 3ix - 2ix + 6i^2 = x^2 - 5ix + 6i^2
Q: How do I simplify the expression?
A: To simplify the expression, you can substitute i^2 = -1. In this case, we can simplify the expression as follows:
f(x) = x^2 - 5ix + 6i^2 = x^2 - 5ix - 6
Q: What is the final answer?
A: The final answer is option C.
Q: What are some related topics?
A: Some related topics include:
- Roots of a polynomial function
- Multiplicity of a root
- Factoring a polynomial function
- Multiplying factors to find a polynomial function
Q: What are some references?
A: Some references include:
- [1] "Algebra" by Michael Artin
- [2] "Polynomial Functions" by Wolfram MathWorld
Q: What is the glossary?
A: The glossary includes the following terms:
- Leading coefficient: The coefficient of the highest degree term in a polynomial function.
- Roots: The values of the variable that make a polynomial function equal to zero.
- Multiplicity: The number of times that a root appears in the factorization of a polynomial function.
- Factoring: The process of expressing a polynomial function as a product of simpler polynomial functions.
- Multiplying factors: The process of multiplying the factors of a polynomial function to find the polynomial function.
Conclusion
In conclusion, we have answered some frequently asked questions about polynomial functions with leading coefficient of 1 and roots 2i and 3i with multiplicity 1. We hope that this article has been helpful in understanding the concept of polynomial functions and how to factor them using the roots.