According to the Fundamental Theorem of Algebra, how many roots exist for the polynomial function?
[tex]f(x)=\left(x^3-3 x+1\right)^2[/tex]
A. 2 roots
B. 3 roots
C. 6 roots
D. 9 roots
Answer (1)
Determine the degree of the polynomial f ( x ) = ( x 3 − 3 x + 1 ) 2 , which is 2 × 3 = 6 .
Apply the Fundamental Theorem of Algebra, which states that a polynomial of degree n has n roots.
Conclude that the polynomial f ( x ) has 6 roots.
The number of roots for the polynomial function is 6 .
Explanation
Understanding the Problem We are given the polynomial function f ( x ) = ( x 3 − 3 x + 1 ) 2 . The Fundamental Theorem of Algebra tells us that a polynomial of degree n has exactly n complex roots, counted with multiplicity. Our goal is to find the number of roots of the given polynomial.
Determining the Degree of the Polynomial First, we need to determine the degree of the polynomial. The expression inside the parentheses, x 3 − 3 x + 1 , is a polynomial of degree 3. Since f ( x ) is the square of this polynomial, we have degree of f ( x ) = 2 × degree of ( x 3 − 3 x + 1 ) = 2 × 3 = 6 Thus, the degree of the polynomial f ( x ) is 6.
Applying the Fundamental Theorem of Algebra Now, we apply the Fundamental Theorem of Algebra. Since the degree of the polynomial f ( x ) is 6, the polynomial has exactly 6 roots, counted with multiplicity.
Final Answer Therefore, according to the Fundamental Theorem of Algebra, the polynomial function f ( x ) = ( x 3 − 3 x + 1 ) 2 has 6 roots.
Examples
The Fundamental Theorem of Algebra is not just an abstract concept; it has practical applications in various fields. For example, in control systems engineering, understanding the roots of a polynomial helps engineers design stable systems. The roots determine the system's response to different inputs, and ensuring these roots have certain properties guarantees the system will behave predictably and safely. Similarly, in signal processing, the roots of polynomials are used to analyze and design filters that remove unwanted noise from signals. Knowing the number and nature of these roots is crucial for effective filter design. In cryptography, polynomials are used to create encryption keys, and the properties of their roots determine the security of the encryption method. Thus, the Fundamental Theorem of Algebra provides a foundation for solving real-world problems in engineering and technology.
