Question

according to the fundamental theorem of algebra, how many roots exist for the polynomial function?

(f(x) = 8x^7 - x^5 + x^3 + 6)

• 3 roots
• 4 roots
• 7 roots
• 8 roots

Explanation:

🆕 New Concept Discovered: Fundamental Theorem of Algebra
The degree of a polynomial determines its total number of complex roots.

Step 1: Identify the polynomial function and its degree

The given polynomial function is:

$$ f(x) = 8x^7 - x^5 + x^3 + 6 $$

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

• \( 8x^7 \) has an exponent of \( 7 \)
• \( -x^5 \) has an exponent of \( 5 \)
• \( x^3 \) has an exponent of \( 3 \)
• \( 6 \) has an exponent of \( 0 \)

The highest exponent is \( 7 \), so the degree of the polynomial is \( 7 \).

Step 2: Apply the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) (where \( n \ge 1 \)) has exactly \( n \) complex roots (including real roots and imaginary roots, counted with their multiplicities).

Since the degree of \( f(x) \) is \( 7 \), the function has exactly \( 7 \) roots.

Answer:

7 roots