Question

according to the fundamental theorem of algebra, which polynomial function has exactly 6 roots?

\\( f(x) = -6x^5 + 10x^2 + 2 \\)
\\( f(x) = 5x^5 + 3x^4 + 12x^3 + 7x^2 - 2x + 10 \\)
\\( f(x) = 8x^6 + x^5 + 4x^3 + x - 5 \\)
\\( f(x) = 7x^6 + 3x^3 + 12 \\)

Explanation:

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

Step 1: Understand the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial function of degree \( n \) (where \( n \geq 1 \)) has exactly \( n \) complex roots (including real roots and any repeated roots).

To find which polynomial function has exactly 6 roots, we need to look for the polynomial with a degree of 6. The degree of a polynomial is the highest exponent of the variable \( x \) in the expression.

Step 2: Analyze the given options

Let's identify the degree of each polynomial option:

1. First option:

$$ f(x) = 4x^3 + 10x^2 + 2 $$

The highest exponent is 3, so the degree is 3. This function has exactly 3 roots.

1. Second option:

$$ f(x) = 5x^5 + 3x^4 + 12x^3 + 7x^2 - 2x + 10 $$

The highest exponent is 5, so the degree is 5. This function has exactly 5 roots.

1. Third option:

$$ f(x) = 8x^4 + x^3 + 4x^2 + x - 5 $$

The highest exponent is 4, so the degree is 4. This function has exactly 4 roots.

1. Fourth option:

$$ f(x) = 7x^6 + 3x^3 + 12 $$

The highest exponent is 6, so the degree is 6. This function has exactly 6 roots.

Answer:

The polynomial function that has exactly 6 roots is:

$$ f(x) = 7x^6 + 3x^3 + 12 $$