According to the Fundamental Theorem of Algebra, how many roots exist for the polynomial function?
$f(x)=4 x^5-3 x$
The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. A significant corollary of this theorem is that every polynomial of degree \(n \ge 1\) has exactly \(n\) roots in the complex plane, when counted with their multiplicity. For the given polynomial function \(f(x) = 4x^5 - 3x\), we first identify its degree. The degree of a polynomial is the highest power of the variable \(x\) present in the expression. Here, the term with the highest power is \(4x^5\), so the degree of this polynomial is \(n = 5\).
By applying the corollary of the Fundamental Theorem of Algebra, which states that a polynomial of degree \(n\) has exactly \(n\) roots in the complex plane, we can directly determine the number of roots for our specific function. Since the degree of the polynomial \(f(x) = 4x^5 - 3x\) is 5, it must have exactly 5 roots in the complex plane, counting multiplicity. Comparing this with the provided options:
The correct option is b.
The Fundamental Theorem of Algebra is a cornerstone in algebra that allows engineers and scientists to understand the behavior of systems. For example, in control systems engineering, the stability of a system is determined by the roots of its characteristic polynomial. If a system is modeled by a 5th-degree polynomial, this theorem guarantees that there are exactly 5 poles, which are critical for analyzing the system's stability and response time. Knowing the total count of these roots ensures that no part of the system's dynamics is overlooked.