How many roots does the following function have (including complex)? \( f(x)=4 x^{5}-6 x^{2}-3 x+7 \) \( \frac{5}{3} \) 6

According to the **Fundamental Theorem of Algebra**, a polynomial of degree \( n \) has exactly \( n \) roots in the complex number system (counting multiplicities). Let's analyze the given function: \[ f(x) = 4x^5 - 6x^2 - 3x + 7 \] - **Degree of the Polynomial:** The highest power of \( x \) is 5, so it's a **5th-degree** polynomial. - **Number of Roots:** Therefore, \( f(x) \) has exactly **5 roots** in the complex plane (this includes real and non-real complex roots). ### Summary: - **Total Roots (including complex):** **5** ### Note on Provided Options: If the options provided were \( \frac{5}{3} \) and **6**, neither of these correctly represents the number of roots for the given polynomial. The accurate number of roots is **5**.