Polynomial Function With 11 Roots Explained By Fundamental Theorem Of Algebra

Jul 7, 2025 by Admin 78 views

According to the Fundamental Theorem of Algebra, Which Polynomial Function Has Exactly 11 Roots?

Understanding the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a cornerstone concept in the field of mathematics, particularly within polynomial equations. This theorem provides a profound insight into the nature of polynomial roots, asserting that every non-constant single-variable polynomial with complex coefficients has at least one complex root. In simpler terms, a complex root can be a real number or a number with an imaginary component. Building on this foundation, a more comprehensive version of the theorem states that a polynomial of degree n, where n is a positive integer, has exactly n complex roots, counted with multiplicity. Multiplicity refers to the number of times a particular root appears as a solution of the polynomial equation. For instance, in the quadratic equation (x - 2)^2 = 0, the root 2 has a multiplicity of 2 because the factor (x - 2) appears twice.

To fully grasp the implications of the Fundamental Theorem of Algebra, it's essential to differentiate between the degree of a polynomial and the number of distinct roots. The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $x^5 + 3x^3 - 2x + 1$, the degree is 5. According to the Fundamental Theorem of Algebra, this polynomial will have exactly 5 roots, considering multiplicity. However, these roots may not all be distinct. Some roots may be repeated, which is where the concept of multiplicity comes into play. For example, consider the polynomial $(x - 1)^3(x + 2)^2$. This polynomial has a degree of 5 (3 from the first factor and 2 from the second). The root 1 has a multiplicity of 3, and the root -2 has a multiplicity of 2. Thus, there are only two distinct roots (1 and -2), but when counted with multiplicity, there are five roots in total (1, 1, 1, -2, -2). Understanding the interplay between the degree of a polynomial and the multiplicity of its roots is crucial for accurately determining the number of roots a polynomial has.

Moreover, the Fundamental Theorem of Algebra has far-reaching implications in various branches of mathematics and related fields. It serves as a foundational principle in complex analysis, where the study of complex functions is paramount. It is also essential in numerical analysis, where algorithms are developed to approximate the roots of polynomials. In engineering and physics, polynomial equations arise frequently in modeling physical systems, and the Fundamental Theorem of Algebra provides the theoretical basis for understanding the solutions to these equations. The theorem is not just an abstract mathematical concept; it has tangible applications in solving real-world problems. For example, in control systems engineering, the stability of a system is often determined by the roots of a characteristic polynomial. The Fundamental Theorem of Algebra guarantees that all the roots of this polynomial can be found, which is crucial for analyzing the system's behavior.

Analyzing the Given Polynomial Functions

To determine which polynomial function has exactly 11 roots according to the Fundamental Theorem of Algebra, we need to analyze the degree of each given polynomial. Remember, the Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex roots, counted with multiplicity. Therefore, we are looking for the polynomial whose degree is 11.

Let's examine each option:

A. $f(x) = (x - 1)(x + 1)^{11}$ This polynomial is a product of two factors. The first factor, $(x - 1)$, has a degree of 1. The second factor, $(x + 1)^{11}$, has a degree of 11. To find the degree of the entire polynomial, we add the degrees of the factors. Thus, the degree of $f(x)$ is $1 + 11 = 12$. According to the Fundamental Theorem of Algebra, this polynomial has 12 roots.

B. $f(x) = (x + 2)^3(x2 - 7x + 3)^4$ This polynomial is also a product of two factors. The first factor, $(x + 2)^3$, has a degree of 3. The second factor, $(x^2 - 7x + 3)^4$, has a degree of $2 * 4 = 8$. To find the degree of the entire polynomial, we add the degrees of the factors. Thus, the degree of $f(x)$ is $3 + 8 = 11$. According to the Fundamental Theorem of Algebra, this polynomial has 11 roots.

C. $f(x) = (x^5 + 7x + 14)^6$ In this case, we have a single factor raised to a power. The factor $(x^5 + 7x + 14)$ has a degree of 5. Since this factor is raised to the power of 6, we multiply the degree of the factor by the exponent. Thus, the degree of $f(x)$ is $5 * 6 = 30$. According to the Fundamental Theorem of Algebra, this polynomial has 30 roots.

D. $f(x) = 11$ This is a constant function. Constant functions have a degree of 0. Therefore, this polynomial has 0 roots.

By analyzing the degree of each polynomial function, we can clearly see that option B, $f(x) = (x + 2)^3(x2 - 7x + 3)^4$, is the only one with a degree of 11. Therefore, according to the Fundamental Theorem of Algebra, this polynomial function has exactly 11 roots.

Detailed Explanation of Option B

Option B, $f(x) = (x + 2)^3(x2 - 7x + 3)^4$, is the correct answer because it is the only polynomial function among the given options that has a degree of 11. To understand why this is the case, let's break down the polynomial into its constituent factors and analyze their degrees individually. The polynomial is composed of two main factors: $(x + 2)^3$ and $(x^2 - 7x + 3)^4$.

The first factor, $(x + 2)^3$, is a cubic expression raised to the power of 3. The term inside the parentheses, $(x + 2)$, is a linear expression with a degree of 1. When this linear expression is raised to the power of 3, the degree of the resulting polynomial is the product of the original degree and the exponent, which in this case is $1 * 3 = 3$. This means that the factor $(x + 2)^3$ contributes a degree of 3 to the overall polynomial. The root of this factor is x = -2, and it has a multiplicity of 3, meaning that the root -2 appears three times as a solution to the polynomial equation.

The second factor, $(x^2 - 7x + 3)^4$, is a quadratic expression raised to the power of 4. The term inside the parentheses, $(x^2 - 7x + 3)$, is a quadratic expression with a degree of 2. When this quadratic expression is raised to the power of 4, the degree of the resulting polynomial is the product of the original degree and the exponent, which in this case is $2 * 4 = 8$. This means that the factor $(x^2 - 7x + 3)^4$ contributes a degree of 8 to the overall polynomial. To find the roots of this factor, we would need to solve the quadratic equation $x^2 - 7x + 3 = 0$. This can be done using the quadratic formula, which gives us two distinct roots. However, since the entire factor is raised to the power of 4, each of these roots has a multiplicity of 4. This means that each root appears four times as a solution to the polynomial equation, contributing a total of 8 roots when counted with multiplicity.

To find the degree of the entire polynomial $f(x)$, we add the degrees of the individual factors. In this case, we add the degree of $(x + 2)^3$, which is 3, to the degree of $(x^2 - 7x + 3)^4$, which is 8. This gives us a total degree of $3 + 8 = 11$. According to the Fundamental Theorem of Algebra, a polynomial of degree 11 has exactly 11 roots, counted with multiplicity. Therefore, the polynomial function $f(x) = (x + 2)^3(x2 - 7x + 3)^4$ has exactly 11 roots.

In summary, option B is the correct answer because the degree of the polynomial $f(x) = (x + 2)^3(x2 - 7x + 3)^4$ is 11. The factor $(x + 2)^3$ contributes 3 roots (all equal to -2), and the factor $(x^2 - 7x + 3)^4$ contributes 8 roots (two distinct roots, each with a multiplicity of 4), for a total of 11 roots. This aligns perfectly with the Fundamental Theorem of Algebra, which guarantees that a polynomial of degree 11 will have exactly 11 roots when counted with multiplicity.

Conclusion

In conclusion, the polynomial function that has exactly 11 roots, according to the Fundamental Theorem of Algebra, is B. $f(x) = (x + 2)^3(x2 - 7x + 3)^4$. This determination is made by understanding that the degree of a polynomial dictates the number of roots it possesses, considering multiplicity. The Fundamental Theorem of Algebra is a pivotal concept in mathematics, providing a foundational understanding of polynomial equations and their solutions. By correctly applying this theorem and analyzing the degrees of the given polynomial functions, we can confidently identify the function with precisely 11 roots.