To determine the number of roots for the polynomial function [tex]\( f(x) = 4x^5 - 3x \)[/tex], we need to consider the degree of the polynomial.
1. Identify the polynomial function: The given polynomial function is [tex]\( f(x) = 4x^5 - 3x \)[/tex].
2. Determine the degree of the polynomial: The degree of a polynomial function is the highest power of the variable [tex]\( x \)[/tex]. In this case, the term with the highest power of [tex]\( x \)[/tex] is [tex]\( 4x^5 \)[/tex]. Therefore, the degree of the polynomial is 5.
3. Apply the Fundamental Theorem of Algebra: The Fundamental Theorem of Algebra states that a polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots in the complex number system (counting multiplicity). Since our polynomial has a degree of 5, it must have exactly 5 roots.
Thus, the number of roots for the polynomial function [tex]\( f(x) = 4x^5 - 3x \)[/tex] is 5.
So, the correct answer is:
- 5 roots
