Question

determine the total number of roots of each polynomial function.

\\(f(x) = 3x^6 + 2x^5 + x^4 - 2x^3\\)

Explanation:

⚡ Using what you learned: Understanding Polynomial Functions

Step 1: Identify the degree of the polynomial

The degree of a polynomial is the highest exponent of the variable \( x \) in its terms.

For the given function:

$$ f(x) = 3x^6 + 2x^5 + x^4 - 2x^3 $$

The highest exponent is \( 6 \), so the degree of the polynomial is \( 6 \).

Step 2: Apply the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) has exactly \( n \) complex roots (including real roots and imaginary roots, counted with multiplicity).

Since the degree of \( f(x) \) is \( 6 \), the total number of roots is \( 6 \).

Answer:

6