If A Polynomial Function $f(x$\] Has Roots $4 - 13i$ And $5$, What Must Be A Factor Of $f(x$\]?A. $(x + (13 - 4i)$\]B. $(x - (13 + 4i)$\]C. $(x + (4 + 13i)$\]D. $(x - (4 + 13i)$\]

Feb 26, 2025 by ADMIN 180 views

**Understanding Polynomial Functions and Their Factors**

Introduction to Polynomial Functions

A polynomial function is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. These functions are widely used in mathematics, physics, and engineering to model real-world phenomena. In this article, we will focus on understanding the relationship between polynomial functions and their factors, specifically in the context of complex roots.

Complex Roots and Factors

When a polynomial function has complex roots, it means that the function can be factored into linear factors, each corresponding to a root of the function. In this case, we are given a polynomial function $f(x)$ with roots $4 - 13i$ and $5$ . We need to determine a factor of $f(x)$ based on these roots.

Understanding Complex Conjugates

In mathematics, complex conjugates are pairs of complex numbers that have the same real part and opposite imaginary parts. For example, the complex conjugate of $4 - 13i$ is $4 + 13i$ . This concept is crucial in understanding the factors of polynomial functions with complex roots.

Factors of Polynomial Functions

A factor of a polynomial function is a polynomial that divides the function without leaving a remainder. In the context of complex roots, a factor of $f(x)$ can be written in the form $(x - r)$ , where $r$ is a root of the function. Since we are given two roots, $4 - 13i$ and $5$ , we can write two factors of $f(x)$ as $(x - (4 - 13i))$ and $(x - 5)$ .

Choosing the Correct Factor

Now, let's analyze the options given in the problem. We need to choose the correct factor of $f(x)$ based on the roots $4 - 13i$ and $5$ . The options are:

A. $(x + (13 - 4i))$ B. $(x - (13 + 4i))$ C. $(x + (4 + 13i))$ D. $(x - (4 + 13i))$

Analyzing Option A

Option A is $(x + (13 - 4i))$ . However, this option does not correspond to the root $4 - 13i$ . The complex conjugate of $4 - 13i$ is $4 + 13i$ , not $13 - 4i$ . Therefore, option A is not the correct factor.

Analyzing Option B

Option B is $(x - (13 + 4i))$ . This option also does not correspond to the root $4 - 13i$ . The complex conjugate of $4 - 13i$ is $4 + 13i$ , not $13 + 4i$ . Therefore, option B is not the correct factor.

Analyzing Option C

Option C is $(x + (4 + 13i))$ . This option corresponds to the complex conjugate of the root $4 - 13i$ , which is $4 + 13i$ . However, the root $5$ is not included in this factor. Therefore, option C is not the correct factor.

Analyzing Option D

Option D is $(x - (4 + 13i))$ . This option corresponds to the complex conjugate of the root $4 - 13i$ , which is $4 + 13i$ . However, the root $5$ is not included in this factor. But, we can write the factor corresponding to the root $5$ as $(x - 5)$ . Therefore, the correct factor of $f(x)$ is the product of these two factors, which is $(x - (4 + 13i))(x - 5)$ .

Conclusion

In conclusion, the correct factor of $f(x)$ is $(x - (4 + 13i))(x - 5)$ . This factor corresponds to the roots $4 - 13i$ and $5$ of the polynomial function $f(x)$ . Therefore, the correct answer is:

D. $(x - (4 + 13i))$

Note that this factor can be written in the form $(x - (4 + 13i))$ , which is the same as option D.

Q: What is a polynomial function?

A: A polynomial function is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents.

Q: What are complex roots and factors?

A: Complex roots are roots of a polynomial function that are complex numbers. Factors of a polynomial function are polynomials that divide the function without leaving a remainder.

Q: What is the relationship between complex roots and factors?

A: Complex roots and factors are related in that each complex root corresponds to a factor of the polynomial function. The factor corresponding to a complex root is in the form $(x - r)$ , where $r$ is the complex root.

Q: What is the concept of complex conjugates?

A: Complex conjugates are pairs of complex numbers that have the same real part and opposite imaginary parts. For example, the complex conjugate of $4 - 13i$ is $4 + 13i$ .

Q: How do complex conjugates relate to factors of polynomial functions?

A: Complex conjugates are used to find the factors of polynomial functions with complex roots. The factor corresponding to a complex root is the product of the complex root and its complex conjugate.

Q: What is the significance of the root $5$ in the problem?

A: The root $5$ is a real root of the polynomial function $f(x)$ . It means that the factor corresponding to this root is $(x - 5)$ .

Q: How do we find the correct factor of $f(x)$ ?

A: To find the correct factor of $f(x)$ , we need to analyze the options given and determine which one corresponds to the roots $4 - 13i$ and $5$ . We can do this by using the concept of complex conjugates and the relationship between complex roots and factors.

Q: What is the correct factor of $f(x)$ ?

A: The correct factor of $f(x)$ is $(x - (4 + 13i))(x - 5)$ .

Q: Why is option D the correct answer?

A: Option D is the correct answer because it corresponds to the complex conjugate of the root $4 - 13i$ , which is $4 + 13i$ , and the root $5$ . The factor $(x - (4 + 13i))$ is the product of the complex conjugate of the root $4 - 13i$ and the factor $(x - 5)$ corresponding to the root $5$ .

Q: What is the significance of the product of the factors?

A: The product of the factors is the correct factor of $f(x)$ . It means that the polynomial function $f(x)$ can be factored into the product of the factors $(x - (4 + 13i))$ and $(x - 5)$ .

Q: How can we use this knowledge in real-world applications?

A: This knowledge can be used in real-world applications such as modeling population growth, chemical reactions, and electrical circuits. Polynomial functions are used to model these phenomena, and understanding the relationship between complex roots and factors is crucial in solving these problems.

Q: What are some common mistakes to avoid when working with polynomial functions and their factors?

A: Some common mistakes to avoid when working with polynomial functions and their factors include:

Not using complex conjugates to find the factors of polynomial functions with complex roots.
Not analyzing the options given to determine the correct factor of the polynomial function.
Not using the product of the factors to find the correct factor of the polynomial function.

Q: How can we practice and improve our understanding of polynomial functions and their factors?

A: We can practice and improve our understanding of polynomial functions and their factors by:

Solving problems and exercises that involve polynomial functions and their factors.
Analyzing and understanding the relationship between complex roots and factors.
Using complex conjugates to find the factors of polynomial functions with complex roots.
Using the product of the factors to find the correct factor of the polynomial function.