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According to the Fundamental Theorem of Algebra, how many roots exist for the polynomial function?

[tex]\[ f(x) = 4x^5 - 3x \][/tex]

A. 1 root
B. 2 roots
C. 4 roots
D. 5 roots

Sagot :

GinnyAnsweridn
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
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