Fundamental Theorem Of Algebra Roots For Polynomial Function F(x)=(x^3-3x+1)^2

Polynomial roots are a cornerstone of algebra, and understanding how to determine the number of roots is crucial for solving polynomial equations. The Fundamental Theorem of Algebra provides a powerful tool for this, stating that a polynomial of degree n has exactly n complex roots, counted with multiplicity. In this comprehensive article, we will delve into the application of this theorem to the polynomial function $f(x) = (x^3 - 3x + 1)^2$, determining the number of roots it possesses. This involves a detailed examination of the polynomial's degree, multiplicity of roots, and the implications of the Fundamental Theorem of Algebra. Before diving into the specifics of the given polynomial, it's essential to grasp the fundamental concepts that underpin the determination of roots. The degree of a polynomial is the highest power of the variable in the polynomial. For instance, in the polynomial $x^4 + 3x^2 - 2x + 1$, the degree is 4. The Fundamental Theorem of Algebra directly links the degree of a polynomial to the number of roots it has. A root of a polynomial function f(x) is a value of x for which f(x) = 0. Roots can be real or complex numbers, and a polynomial of degree n will have n roots in the complex number system. These roots may not all be distinct; some roots may be repeated. This repetition is known as multiplicity. If a root appears k times, it has a multiplicity of k. For example, in the equation $(x-2)^3 = 0$, the root x = 2 has a multiplicity of 3. Understanding multiplicity is critical because the Fundamental Theorem of Algebra counts roots with their respective multiplicities. This means a polynomial of degree n will have n roots when each root is counted as many times as its multiplicity indicates. This theorem provides a powerful foundation for analyzing polynomial functions and their solutions.

Deconstructing the Polynomial Function $f(x) = (x^3 - 3x + 1)^2$

To determine the number of roots for the polynomial function $f(x) = (x^3 - 3x + 1)^2$, we must first ascertain the degree of the polynomial. This involves understanding how the given expression transforms into its standard polynomial form. The function is presented as a squared expression, where the base is another polynomial, $x^3 - 3x + 1$. This inner polynomial is a cubic polynomial, meaning its degree is 3. The degree of a polynomial is the highest power of the variable x. When we square the cubic polynomial $(x^3 - 3x + 1)$, we are essentially multiplying it by itself: $(x^3 - 3x + 1)(x^3 - 3x + 1)$. This multiplication process will increase the degree of the resulting polynomial. To find the degree of the squared polynomial, we can use the property that the degree of the product of two polynomials is the sum of their individual degrees. In this case, we have a polynomial of degree 3 multiplied by another polynomial of degree 3. Therefore, the resulting polynomial will have a degree of 3 + 3 = 6. Alternatively, we can consider the highest power term in the original cubic polynomial, which is $x^3$. When we square this term, we get $(x^3)^2 = x^6$. This confirms that the degree of the polynomial $f(x) = (x^3 - 3x + 1)^2$ is indeed 6. Understanding the degree is paramount because it directly relates to the number of roots the polynomial possesses, as stated by the Fundamental Theorem of Algebra. Once we have established the degree, we can confidently apply the theorem to determine the total number of roots, considering both real and complex roots, and accounting for multiplicities. The degree of a polynomial is a crucial indicator of its complexity and behavior, particularly in terms of its roots. Higher-degree polynomials can exhibit more complex root structures, including multiple real and complex roots, and the Fundamental Theorem of Algebra provides a framework for understanding this complexity.

Applying the Fundamental Theorem of Algebra

Now that we have established that the polynomial function $f(x) = (x^3 - 3x + 1)^2$ has a degree of 6, we can directly apply the Fundamental Theorem of Algebra to determine the number of roots. The theorem states that a polynomial of degree n has exactly n complex roots, counted with multiplicity. This means that our polynomial, with a degree of 6, will have precisely 6 roots in the complex number system. These roots can be real or complex, and they may not all be distinct. Some roots may be repeated, which is referred to as their multiplicity. To fully understand the implications of this, let's consider the structure of the function. The polynomial is given as a square of another polynomial, $(x^3 - 3x + 1)^2$. This squaring operation has a significant impact on the roots. If a value r is a root of the cubic polynomial $x^3 - 3x + 1$, then $(x^3 - 3x + 1)$ will have a factor of $(x - r)$. When we square the polynomial, we are essentially squaring this factor, resulting in $(x - r)^2$. This means that the root r will now have a multiplicity of 2 in the squared polynomial. This is a crucial observation because it tells us that each root of the cubic polynomial will appear twice in the roots of the sixth-degree polynomial. The cubic polynomial $x^3 - 3x + 1$ itself has 3 roots, according to the Fundamental Theorem of Algebra. These roots may be real or complex, and they may not be easily determined analytically. However, regardless of their nature, each of these 3 roots will be duplicated when the polynomial is squared. Therefore, the sixth-degree polynomial $f(x) = (x^3 - 3x + 1)^2$ will have 6 roots, with each root of the cubic polynomial having a multiplicity of 2. This concept of multiplicity is essential for correctly counting the roots and understanding the complete solution set of the polynomial equation. The Fundamental Theorem of Algebra guarantees the existence of these 6 roots, providing a definitive answer to the question of how many roots exist for the given polynomial function.

Conclusion: Determining the Number of Roots

In conclusion, by applying the Fundamental Theorem of Algebra to the polynomial function $f(x) = (x^3 - 3x + 1)^2$, we have definitively determined the number of roots. The process involved several key steps, starting with identifying the degree of the polynomial. We recognized that the given function is the square of a cubic polynomial. This means that when expanded, the polynomial will have a degree of 6. Understanding the degree is crucial because the Fundamental Theorem of Algebra directly links the degree of a polynomial to the number of roots it possesses. The theorem states that a polynomial of degree n has exactly n complex roots, counted with multiplicity. Therefore, since our polynomial has a degree of 6, it has 6 roots. These roots may be real or complex, and they may not all be distinct. The multiplicity of roots plays a significant role in this count. The fact that the polynomial is a square of another polynomial implies that each root of the inner cubic polynomial will appear twice in the roots of the sixth-degree polynomial. This further reinforces the conclusion that there are 6 roots, with each root having a multiplicity of 2. By carefully considering the degree of the polynomial and the implications of the Fundamental Theorem of Algebra, we have arrived at a clear and concise answer. The polynomial function $f(x) = (x^3 - 3x + 1)^2$ has 6 roots. This understanding is crucial for solving polynomial equations and analyzing the behavior of polynomial functions. The Fundamental Theorem of Algebra provides a powerful and fundamental tool for understanding the nature and number of roots, making it a cornerstone of algebraic analysis. Therefore, the correct answer is C. 6 roots.